family of positive integers that occur as coefficients in the binomial theorem
Wikipedia
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Binomial coefficient - Wikipedia
2 weeks ago - {\displaystyle 0\leq t<s,} series multisection gives the following identity for the sum of binomial coefficients:
History and notationDefinition and interpretationsComputing the value of binomial coefficientsPascal's triangleCombinatorics and statisticsBinomial coefficients as polynomialsIdentities involving binomial coefficientsGenerating functionsDivisibility propertiesBounds and asymptotic formulasGeneralizations
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The most comprehensive list I know of is H.W. Gould's Combinatorial Identities. It is available directly from him if you contact him. He also has some pdf documents available for download from his web site. Although he says they do "NOT replace [Combinatorial Identities] which remains in print with supplements," they still contain many more binomial identities even than in Concrete Mathematics. In general, Gould's work is a great resource for this sort of thing; he has spent much of his career collecting and proving combinatorial identities.
Added: Another useful reference is John Riordan's Combinatorial Identities. It's hard to pick one of its 250 pages at random and not find at least one binomial coefficient identity there. Unfortunately, the identities are not always organized in a way that makes it easy to find what you are looking for. Still it's a good resource.
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If you do possess Concrete Mathematics, check out 5.1 Basic Identities and 5.2 Basic Practice. The list can be found at P174 Table 174 The top ten binomial coefficient identities.
Videos
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Binomial Coefficients Identity with Induction - YouTube
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[Discrete Mathematics] Section 6.7. Binomial Coefficients and ...
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Proof: Recursive Identity for Binomial Coefficients | Combinatorics ...
Math 2200: Section 11.2 - Binomial Coefficients and ...
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6.4 Binomial Coefficients and Identities - YouTube
Discrete Math - 6.4.1 The Binomial Theorem
UCSD Mathematics
mathweb.ucsd.edu › ~gptesler › 184a › slides › 184a_ch4slides_17-handout.pdf pdf
Chapter 3.3, 4.1, 4.3. Binomial Coefficient Identities Prof. Tesler Math 184A
expressing binomial coefficients in terms of factorials.
Wm
math.wm.edu › ~shij › putnam › bino.pdf pdf
Binomial identities, binomial coefficients, and binomial theorem
Pascal’s triangle is a geometric arrangement of the binomial coefficients in a triangle. Pascal’s · triangle can be constructed using Pascal’s rule (or addition formula), which states that ... for this sequence. The generating function for the sequence (fn−1) is X ·f(X) and that of (fn−2) is · X2 · f(X). From the recurrence relation, we therefore see that the power series Xf(X) + X2f(X) agrees with f(X) except for the first two coefficients.
ProofWiki
proofwiki.org › wiki › Properties_of_Binomial_Coefficients
Properties of Binomial Coefficients - ProofWiki
16 Increasing Alternating Sum of Binomial Coefficients · 17 Chu-Vandermonde Identity · 18 Sum of Squares of Binomial Coefficients · 19 Binomial Coefficient: $\left({-1}\right)^n \dbinom {-n} {k - 1} = \left({-1}\right)^k \dbinom {-k} {n - 1}$ 20 Binomial Coefficient as Infinite Product ·
Wolfram MathWorld
mathworld.wolfram.com › BinomialCoefficient.html
Binomial Coefficient -- from Wolfram MathWorld
May 24, 2024 - Apéry Number, Balanced Binomial Coefficient, Ballot Problem, Bernoulli Triangle, Binomial, Binomial Distribution, Binomial Identity, Binomial Sums, Binomial Theorem, Central Binomial Coefficient, Choose, Christmas Stocking Theorem, Chu-Vandermonde Identity, Combination, Deficiency, Erdős Squarefree Conjecture, Exceptional Binomial Coefficient, Factorial, Fibonomial Coefficient, Gamma Function, Good Binomial Coefficient, k-Subset, Kings Problem, Klee's Identity, Lah Number, Multichoose, Multinomial Coefficient, Pascal's Formula, Permutation, q-Binomial Coefficient, Roman Coefficient, Sárkőzy's Theorem, Stanley's Identity, Star of David Theorem, Stolarsky-Harborth Constant, Strehl Identities, Székely Identity, Wolstenholme's Theorem Explore this topic in the MathWorld classroom
Columbia University
cs.columbia.edu › ~cs4205 › files › CM4.pdf pdf
Chapter 4 Binomial Coefficients 4.1 Binomial Coefficient Identities
4.1 Binomial Coefficient Identities · 4.2 Binomial Inversion Operation · 4.3 Applications to Statistics · 4.4 The Catalan Recurrence · 1 · 2 · Chapter 4 · Binomial Coefficients · 4.1 · BINOMIAL COEFF IDENTITIES · Table 4.1.1 · Section 4.1 · Binomial Coeff Identities ·
Math
fq.math.ca › Papers1 › 49-1 › shephard.pdf pdf
MULTIPLICATIVE IDENTITIES FOR BINOMIAL COEFFICIENTS G. C. SHEPHARD
To prove (10) we insert the expressions for the binomial coefficients in terms of factorials. For · each term r, the term r! occurs in the denominators on the the left of (11) ... x = y and then altering the signs of all its labels, the resulting identity is trivial in that all its
Whitman College
whitman.edu › mathematics › cgt_online › book › section01.03.html
1.3 Binomial coefficients
You may know, for example, that the entries in Pascal's Triangle are the coefficients of the polynomial produced by raising a binomial to an integer power. For example, $\ds (x+y)^3=1\cdot x^3+3\cdot x^2y+ 3\cdot xy^2+1\cdot y^3$, and the coefficients 1, 3, 3, 1 form row three of Pascal's Triangle.
MIT Mathematics
math.mit.edu › research › highschool › primes › materials › 2021 › May › 4-1-Pham.pdf pdf
The binomial theorem and related identities Duy Pham Mentor: Eli Garcia
and related identities · Duy Pham · Mentor: Eli Garcia · Table of contents · Binomial theorem · The pascal’s triangle · Binomial coefficient · Generalized binomial theorem · Trinomial theorem · Multinomial theorem · Vandermonde’s identity · Common mistake ·
University of Waterloo
cs.uwaterloo.ca › journals › JIS › VOL26 › Batir › batir26.pdf pdf
23 11 Article 23.4.4 Journal of Integer Sequences, Vol. 26 (2023), 2 3 6 1 47
[6] H. W. Gould, Combinatorial identities: A standardized set of tables listing 500 binomial · coefficient summations by Henry Wadsworth Gould, Morgantown, 1972.
Wolfram MathWorld
mathworld.wolfram.com › BinomialSums.html
Binomial Sums -- from Wolfram MathWorld
September 27, 2007 - The identity holds true not only for and , but also for any quadratic polynomial of the form . Sinyor et al. (2001) give the strange sum · Apéry Number, Binomial, Binomial Coefficient, Central Binomial Coefficient, Christmas Stocking Theorem, Franel Number, Hypergeometric Identity, Hypergeometric Series, Idempotent Number, Jonah Formula Klee's Identity, Lucas Correspondence Theorem, Married Couples Problem, Morley's Formula, Nexus Number, Pascal's Formula, Schmidt's Problem, Stanley's Identity, Star of David Theorem, Strehl Identities, Székely Identity, Waring Formula, Worpitzky's Identity
Wolfram MathWorld
mathworld.wolfram.com › BinomialIdentity.html
Binomial Identity -- from Wolfram MathWorld
May 2, 2000 - for . Abel (1826) gave a host of such identities (Riordan 1979, Roman 1984), some of which include · (Abel 1826, Riordan 1979, p. 18; Roman 1984, pp. 30 and 73), and ... Abel's Binomial Theorem, Abel Polynomial, Binomial, Binomial Coefficient, Dilcher's Formula, q-Abel's Theorem
Open Math Books
discrete.openmathbooks.org › dmoi3 › sec_comb-proofs.html
Combinatorial Proofs
Find an expression for the answer which is the difference of two binomial coefficients. ... Generalize the above to state and prove a binomial identity using a combinatorial proof.
ResearchGate
researchgate.net › publication › 370101590_On_Some_Binomial_Coefficient_Identities_with_Applications
(PDF) On Some Binomial Coefficient Identities with Applications
April 1, 2023 - Combinatorial identities. A standardized set of tables listing 500 binomial coefficient summations.
University of Houston
math.uh.edu › ~pwalker › 3336Sp21Sec6.4Slides.pdf pdf
© 2019 McGraw-Hill Education Binomial Coefficients and Identities Section 6.4
So, the coefficient of x3 is 1. ... To obtain x2y, an x must be chosen from two of the sums and a y from the other. There are ... To obtain xy2, an x must be chosen from of the sums and a y from the other two . ... To obtain y3 , a y must be chosen from each of the sums. There is only one way to do ... We have used a counting argument to show that (x + y)3 = x3 + 3x2y + 3x y2 + y3 . Next we present the binomial theorem gives the coefficients of the terms in the
OpenMathBooks
discrete.openmathbooks.org › dmoi2 › sec_counting-binom.html
Binomial Coefficients
We still need the coefficients of \(x^3y^2\) and \(x^2y^3\text{.}\) In both cases, we need to pick exactly 3 of the 5 binomials to contribute one variable, the other two to contribute the other. Wait. This sounds familiar. We have 5 things, each can be one of two things, and we need a total of 3 of one of them.
Fiveable
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Binomial Coefficients and Identities | Discrete Mathematics Class Notes | Fiveable
August 22, 2025 - Mastering binomial coefficients opens doors to understanding advanced topics in discrete math. We'll explore their properties, identities, and applications, seeing how they connect to other concepts in counting and probability.
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In terms of hypergeometric series, the sum is _3F_2(-n, 1+n, 1/2;1,3/2;1) · and the identity is a special case of Saalschütz's theorem (also called the Pfaff-Saalschütz theorem), one of the standard hypergeometric series identities. · A more general identity, also a special case of Saalschütz's theorem, is · \sum_{k=0}^n (-1)^k\frac{a}{a+k}\binom{n+k+b}{n-k}\binom{2k+b}{k} · = \binom{n+b-a}{n}\biggm/\binom{n+a}{n}. · The O.P.'s identity is the case a=1/2, b=0 · .
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Use \binom{n+k}{k}\binom{n}k · in the sum. Define the functions · F(n,k)=(-1)^k\frac{2n+1}{2k+1}\binom{n+k}k\binom{n}{k}, \qquad · G(n,k)=2\cdot(-1)^{k-1}\binom{n+k}{k-1}\binom{n}{k-1}. · Then F(n+1,k)-F(n,k)=G(n,k+1)-G(n,k) · . Sum over all integers k · to obtain f(n+1)-f(n)=0 · where f(n)=\sum_kF(n,k) · is your sum. Since f(0)=1 · , the identity follows. · This method is called the Wilf-Zeilberger technique of summation routine.