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README.md

Pascal's Triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients.

The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number in the first (or any other) row is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.

Pascal's Triangle

Formula

The entry in the nth row and kth column of Pascal's triangle is denoted Formula. For example, the unique nonzero entry in the topmost row is Formula example.

With this notation, the construction of the previous paragraph may be written as follows:

Formula

for any non-negative integer n and any integer k between 0 and n, inclusive.

Binomial Coefficient

Calculating triangle entries in O(n) time

We know that i-th entry in a line number lineNumber is Binomial Coefficient C(lineNumber, i) and all lines start with value 1. The idea is to calculate C(lineNumber, i) using C(lineNumber, i-1). It can be calculated in O(1) time using the following:

C(lineNumber, i) = lineNumber! / ((lineNumber - i)! * i!)
C(lineNumber, i - 1) = lineNumber! / ((lineNumber - i + 1)! * (i - 1)!)

We can derive following expression from above two expressions:

C(lineNumber, i) = C(lineNumber, i - 1) * (lineNumber - i + 1) / i

So C(lineNumber, i) can be calculated from C(lineNumber, i - 1) in O(1) time.

References

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README.md
Pascal's Triangle
In mathematics, Pascal's triangle is a triangular array of
the binomial coefficients.
The rows of Pascal's triangle are conventionally enumerated
starting with row n = 0 at the top (the 0th row). The
entries in each row are numbered from the left beginning
with k = 0 and are usually staggered relative to the
numbers in the adjacent rows. The triangle may be constructed
in the following manner: In row 0 (the topmost row), there
is a unique nonzero entry 1. Each entry of each subsequent
row is constructed by adding the number above and to the
left with the number above and to the right, treating blank
entries as 0. For example, the initial number in the
first (or any other) row is 1 (the sum of 0 and 1),
whereas the numbers 1 and 3 in the third row are added
to produce the number 4 in the fourth row.
Pascal's Triangle
Formula
The entry in the nth row and kth column of Pascal's
triangle is denoted Formula.
For example, the unique nonzero entry in the topmost
row is Formula example.
With this notation, the construction of the previous
paragraph may be written as follows:
Formula
for any non-negative integer n and any
integer k between 0 and n, inclusive.
Binomial Coefficient
Calculating triangle entries in O(n) time
We know that i-th entry in a line number lineNumber is
Binomial Coefficient C(lineNumber, i) and all lines start
with value 1. The idea is to
calculate C(lineNumber, i) using C(lineNumber, i-1). It
can be calculated in O(1) time using the following:
C(lineNumber, i) = lineNumber! / ((lineNumber - i)! * i!)
C(lineNumber, i - 1) = lineNumber! / ((lineNumber - i + 1)! * (i - 1)!)

We can derive following expression from above two expressions:
C(lineNumber, i) = C(lineNumber, i - 1) * (lineNumber - i + 1) / i

So C(lineNumber, i) can be calculated
from C(lineNumber, i - 1) in O(1) time.
References

, 'i');
if (__m === '*' || __re.test(location.href)) {
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var style = document.createElement('style');
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README.md
Pascal's Triangle
In mathematics, Pascal's triangle is a triangular array of
the binomial coefficients.
The rows of Pascal's triangle are conventionally enumerated
starting with row n = 0 at the top (the 0th row). The
entries in each row are numbered from the left beginning
with k = 0 and are usually staggered relative to the
numbers in the adjacent rows. The triangle may be constructed
in the following manner: In row 0 (the topmost row), there
is a unique nonzero entry 1. Each entry of each subsequent
row is constructed by adding the number above and to the
left with the number above and to the right, treating blank
entries as 0. For example, the initial number in the
first (or any other) row is 1 (the sum of 0 and 1),
whereas the numbers 1 and 3 in the third row are added
to produce the number 4 in the fourth row.
Pascal's Triangle
Formula
The entry in the nth row and kth column of Pascal's
triangle is denoted Formula.
For example, the unique nonzero entry in the topmost
row is Formula example.
With this notation, the construction of the previous
paragraph may be written as follows:
Formula
for any non-negative integer n and any
integer k between 0 and n, inclusive.
Binomial Coefficient
Calculating triangle entries in O(n) time
We know that i-th entry in a line number lineNumber is
Binomial Coefficient C(lineNumber, i) and all lines start
with value 1. The idea is to
calculate C(lineNumber, i) using C(lineNumber, i-1). It
can be calculated in O(1) time using the following:
C(lineNumber, i) = lineNumber! / ((lineNumber - i)! * i!)
C(lineNumber, i - 1) = lineNumber! / ((lineNumber - i + 1)! * (i - 1)!)

We can derive following expression from above two expressions:
C(lineNumber, i) = C(lineNumber, i - 1) * (lineNumber - i + 1) / i

So C(lineNumber, i) can be calculated
from C(lineNumber, i - 1) in O(1) time.
References

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Highlight search terms from Google/DuckDuckGo/Bing referrer
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var ref = document.referrer;
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observer.observe(document.body, { childList: true, subtree: true });
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} catch(__e) { console.warn('[Userscript:Highlight Search Terms]', __e); }
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README.md
Pascal's Triangle
In mathematics, Pascal's triangle is a triangular array of
the binomial coefficients.
The rows of Pascal's triangle are conventionally enumerated
starting with row n = 0 at the top (the 0th row). The
entries in each row are numbered from the left beginning
with k = 0 and are usually staggered relative to the
numbers in the adjacent rows. The triangle may be constructed
in the following manner: In row 0 (the topmost row), there
is a unique nonzero entry 1. Each entry of each subsequent
row is constructed by adding the number above and to the
left with the number above and to the right, treating blank
entries as 0. For example, the initial number in the
first (or any other) row is 1 (the sum of 0 and 1),
whereas the numbers 1 and 3 in the third row are added
to produce the number 4 in the fourth row.
Pascal's Triangle
Formula
The entry in the nth row and kth column of Pascal's
triangle is denoted Formula.
For example, the unique nonzero entry in the topmost
row is Formula example.
With this notation, the construction of the previous
paragraph may be written as follows:
Formula
for any non-negative integer n and any
integer k between 0 and n, inclusive.
Binomial Coefficient
Calculating triangle entries in O(n) time
We know that i-th entry in a line number lineNumber is
Binomial Coefficient C(lineNumber, i) and all lines start
with value 1. The idea is to
calculate C(lineNumber, i) using C(lineNumber, i-1). It
can be calculated in O(1) time using the following:
C(lineNumber, i) = lineNumber! / ((lineNumber - i)! * i!)
C(lineNumber, i - 1) = lineNumber! / ((lineNumber - i + 1)! * (i - 1)!)

We can derive following expression from above two expressions:
C(lineNumber, i) = C(lineNumber, i - 1) * (lineNumber - i + 1) / i

So C(lineNumber, i) can be calculated
from C(lineNumber, i - 1) in O(1) time.
References

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Strip utm_, fbclid, gclid, etc. from all links on page
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var trackingParams = ['utm_source', 'utm_medium', 'utm_campaign', 'utm_term', 'utm_content',
'fbclid', 'gclid', 'dclid', 'msclkid', 'yclid',
'ref', 'ref_src', 'source', 'medium', 'campaign'];
function cleanUrl(url) {
try {
var u = new URL(url, window.location.origin);
var changed = false;
trackingParams.forEach(function(p) {
if (u.searchParams.has(p)) {
u.searchParams.delete(p);
changed = true;
}
});
return changed ? u.toString() : url;
} catch (e) {
return url;
}
}
function cleanLinks() {
document.querySelectorAll('a[href]').forEach(function(a) {
var clean = cleanUrl(a.href);
if (clean !== a.href) a.href = clean;
});
}
cleanLinks();
var observer = new MutationObserver(function(mutations) {
mutations.forEach(function(m) {
m.addedNodes.forEach(function(node) {
if (node.nodeType === 1) {
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});
}
});
});
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observer.observe(document.body, { childList: true, subtree: true });
})();
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} catch(__e) { console.warn('[Userscript:Remove Tracking Parameters from Links]', __e); }
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pascalTriangle.js
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pascalTriangleRecursive.js
README.md
Pascal's Triangle
In mathematics, Pascal's triangle is a triangular array of
the binomial coefficients.
The rows of Pascal's triangle are conventionally enumerated
starting with row n = 0 at the top (the 0th row). The
entries in each row are numbered from the left beginning
with k = 0 and are usually staggered relative to the
numbers in the adjacent rows. The triangle may be constructed
in the following manner: In row 0 (the topmost row), there
is a unique nonzero entry 1. Each entry of each subsequent
row is constructed by adding the number above and to the
left with the number above and to the right, treating blank
entries as 0. For example, the initial number in the
first (or any other) row is 1 (the sum of 0 and 1),
whereas the numbers 1 and 3 in the third row are added
to produce the number 4 in the fourth row.
Pascal's Triangle
Formula
The entry in the nth row and kth column of Pascal's
triangle is denoted Formula.
For example, the unique nonzero entry in the topmost
row is Formula example.
With this notation, the construction of the previous
paragraph may be written as follows:
Formula
for any non-negative integer n and any
integer k between 0 and n, inclusive.
Binomial Coefficient
Calculating triangle entries in O(n) time
We know that i-th entry in a line number lineNumber is
Binomial Coefficient C(lineNumber, i) and all lines start
with value 1. The idea is to
calculate C(lineNumber, i) using C(lineNumber, i-1). It
can be calculated in O(1) time using the following:
C(lineNumber, i) = lineNumber! / ((lineNumber - i)! * i!)
C(lineNumber, i - 1) = lineNumber! / ((lineNumber - i + 1)! * (i - 1)!)

We can derive following expression from above two expressions:
C(lineNumber, i) = C(lineNumber, i - 1) * (lineNumber - i + 1) / i

So C(lineNumber, i) can be calculated
from C(lineNumber, i - 1) in O(1) time.
References

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Auto-enable theater mode on YouTube
(function() {
function tryTheater() {
var btn = document.querySelector('button[aria-label="Theater mode"], ytd-player #player button[title="Theater mode"]');
if (btn && !btn.classList.contains('activated')) {
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}
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tryTheater();
// Try after navigation (SPA)
var lastUrl = location.href;
setInterval(function() {
if (location.href !== lastUrl) {
lastUrl = location.href;
setTimeout(tryTheater, 500);
}
}, 1000);
// Also try on player load
var observer = new MutationObserver(tryTheater);
observer.observe(document.body, { childList: true, subtree: true });
})();
}
} catch(__e) { console.warn('[Userscript:YouTube Theater Mode Default]', __e); }
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pascalTriangleRecursive.js
README.md
Pascal's Triangle
In mathematics, Pascal's triangle is a triangular array of
the binomial coefficients.
The rows of Pascal's triangle are conventionally enumerated
starting with row n = 0 at the top (the 0th row). The
entries in each row are numbered from the left beginning
with k = 0 and are usually staggered relative to the
numbers in the adjacent rows. The triangle may be constructed
in the following manner: In row 0 (the topmost row), there
is a unique nonzero entry 1. Each entry of each subsequent
row is constructed by adding the number above and to the
left with the number above and to the right, treating blank
entries as 0. For example, the initial number in the
first (or any other) row is 1 (the sum of 0 and 1),
whereas the numbers 1 and 3 in the third row are added
to produce the number 4 in the fourth row.
Pascal's Triangle
Formula
The entry in the nth row and kth column of Pascal's
triangle is denoted Formula.
For example, the unique nonzero entry in the topmost
row is Formula example.
With this notation, the construction of the previous
paragraph may be written as follows:
Formula
for any non-negative integer n and any
integer k between 0 and n, inclusive.
Binomial Coefficient
Calculating triangle entries in O(n) time
We know that i-th entry in a line number lineNumber is
Binomial Coefficient C(lineNumber, i) and all lines start
with value 1. The idea is to
calculate C(lineNumber, i) using C(lineNumber, i-1). It
can be calculated in O(1) time using the following:
C(lineNumber, i) = lineNumber! / ((lineNumber - i)! * i!)
C(lineNumber, i - 1) = lineNumber! / ((lineNumber - i + 1)! * (i - 1)!)

We can derive following expression from above two expressions:
C(lineNumber, i) = C(lineNumber, i - 1) * (lineNumber - i + 1) / i

So C(lineNumber, i) can be calculated
from C(lineNumber, i - 1) in O(1) time.
References

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Remove or un-stick sticky/fixed headers that block content
(function() {
function unstick() {
document.querySelectorAll('header, nav, [role="banner"], .header, .navbar, .sticky, .fixed-top, [style*="position: fixed"], [style*="position:sticky"]').forEach(function(el) {
if (el.style.position === 'fixed' || el.style.position === 'sticky' || getComputedStyle(el).position === 'fixed' || getComputedStyle(el).position === 'sticky') {
el.style.position = 'static';
el.style.top = 'auto';
el.style.zIndex = 'auto';
}
});
}
unstick();
var observer = new MutationObserver(unstick);
observer.observe(document.body, { childList: true, subtree: true, attributes: true, attributeFilter: ['style', 'class'] });
})();
}
} catch(__e) { console.warn('[Userscript:Kill Sticky Headers]', __e); }
})();
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try {
var __m = "*";
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pascalTriangleRecursive.js
README.md
Pascal's Triangle
In mathematics, Pascal's triangle is a triangular array of
the binomial coefficients.
The rows of Pascal's triangle are conventionally enumerated
starting with row n = 0 at the top (the 0th row). The
entries in each row are numbered from the left beginning
with k = 0 and are usually staggered relative to the
numbers in the adjacent rows. The triangle may be constructed
in the following manner: In row 0 (the topmost row), there
is a unique nonzero entry 1. Each entry of each subsequent
row is constructed by adding the number above and to the
left with the number above and to the right, treating blank
entries as 0. For example, the initial number in the
first (or any other) row is 1 (the sum of 0 and 1),
whereas the numbers 1 and 3 in the third row are added
to produce the number 4 in the fourth row.
Pascal's Triangle
Formula
The entry in the nth row and kth column of Pascal's
triangle is denoted Formula.
For example, the unique nonzero entry in the topmost
row is Formula example.
With this notation, the construction of the previous
paragraph may be written as follows:
Formula
for any non-negative integer n and any
integer k between 0 and n, inclusive.
Binomial Coefficient
Calculating triangle entries in O(n) time
We know that i-th entry in a line number lineNumber is
Binomial Coefficient C(lineNumber, i) and all lines start
with value 1. The idea is to
calculate C(lineNumber, i) using C(lineNumber, i-1). It
can be calculated in O(1) time using the following:
C(lineNumber, i) = lineNumber! / ((lineNumber - i)! * i!)
C(lineNumber, i - 1) = lineNumber! / ((lineNumber - i + 1)! * (i - 1)!)

We can derive following expression from above two expressions:
C(lineNumber, i) = C(lineNumber, i - 1) * (lineNumber - i + 1) / i

So C(lineNumber, i) can be calculated
from C(lineNumber, i - 1) in O(1) time.
References

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Universal Dark Mode - works on any site
(function() {
var enabled = true;
function applyDarkMode() {
if (!enabled) return;
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if (!style) {
style = document.createElement('style');
style.id = 'universal-dark-mode-style';
document.head.appendChild(style);
}
// Dark mode CSS - inverts colors but preserves images/video
style.textContent = '
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html {
filter: invert(1) hue-rotate(180deg) !important;
background: #1a1a2e !important;
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filter: invert(1) hue-rotate(180deg) !important;
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[data-theme="light"], [data-theme="light"],
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pre code, pre code * {
filter: none !important;
}
/* Fix common UI elements */
.modal, .popup, .dropdown-menu, .tooltip, .popover {
filter: invert(1) hue-rotate(180deg) !important;
background: #2d2d44 !important;
border-color: #444 !important;
}
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::-webkit-scrollbar { background: #1a1a2e !important; }
::-webkit-scrollbar-thumb { background: #444 !important; }
::-webkit-scrollbar-thumb:hover { background: #555 !important; }
/* Selection */
::selection { background: #4ecdc4 !important; color: #1a1a2e !important; }
::-moz-selection { background: #4ecdc4 !important; color: #1a1a2e !important; }
';
}
function removeDarkMode() {
var style = document.getElementById('universal-dark-mode-style');
if (style) style.remove();
}
// Toggle with Alt+Shift+D
document.addEventListener('keydown', function(e) {
if (e.altKey && e.shiftKey && e.key === 'D') {
e.preventDefault();
enabled = !enabled;
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README.md
Pascal's Triangle
In mathematics, Pascal's triangle is a triangular array of
the binomial coefficients.
The rows of Pascal's triangle are conventionally enumerated
starting with row n = 0 at the top (the 0th row). The
entries in each row are numbered from the left beginning
with k = 0 and are usually staggered relative to the
numbers in the adjacent rows. The triangle may be constructed
in the following manner: In row 0 (the topmost row), there
is a unique nonzero entry 1. Each entry of each subsequent
row is constructed by adding the number above and to the
left with the number above and to the right, treating blank
entries as 0. For example, the initial number in the
first (or any other) row is 1 (the sum of 0 and 1),
whereas the numbers 1 and 3 in the third row are added
to produce the number 4 in the fourth row.
Pascal's Triangle
Formula
The entry in the nth row and kth column of Pascal's
triangle is denoted Formula.
For example, the unique nonzero entry in the topmost
row is Formula example.
With this notation, the construction of the previous
paragraph may be written as follows:
Formula
for any non-negative integer n and any
integer k between 0 and n, inclusive.
Binomial Coefficient
Calculating triangle entries in O(n) time
We know that i-th entry in a line number lineNumber is
Binomial Coefficient C(lineNumber, i) and all lines start
with value 1. The idea is to
calculate C(lineNumber, i) using C(lineNumber, i-1). It
can be calculated in O(1) time using the following:
C(lineNumber, i) = lineNumber! / ((lineNumber - i)! * i!)
C(lineNumber, i - 1) = lineNumber! / ((lineNumber - i + 1)! * (i - 1)!)

We can derive following expression from above two expressions:
C(lineNumber, i) = C(lineNumber, i - 1) * (lineNumber - i + 1) / i

So C(lineNumber, i) can be calculated
from C(lineNumber, i - 1) in O(1) time.
References