Combinatorial and integer sequences

Part of the Fungrim Identities reference — 64 identities for combinatorial and integer sequences.

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Bell numbers (1)
Bernoulli numbers and polynomials (9)
Fibonacci numbers (36)
Integer sequences (8)
Partition function (6)
Stirling numbers (4)

Bell numbers

$\mathrm{BellNumber}(n)=\frac{2\Im(\int_{0}^{\pi}\!\sin(nx)\exp(\exp(\exp(\imaginaryI x)))\, \mathrm{d}x)n!}{\exponentialE\pi}$

Holds when $n\in\N^*$ . Used by the Compute Engine for simplification. Reference: arxiv.org f4e249 · Fungrim entry ↗

Bernoulli numbers and polynomials

$\mathrm{BernoulliPolynomial}(n, \frac{1}{2})=(2^{1-n}-1)\mathrm{BernoulliB}(n)$

Holds when $n\in\N$ . Symbols: BernoulliB — Bernoulli number; BernoulliPolynomial — Bernoulli polynomial. Used by the Compute Engine for simplification. 03ee0b · Fungrim entry ↗

$\mathrm{BernoulliB}(2n)=\frac{2\Zeta(2n)(2n)!\times(-1)^{n+1}}{(2\pi)^{2n}}$

Holds when $n\in\N^*$ . Symbols: BernoulliB — Bernoulli number. Used by the Compute Engine for simplification. 14ecc4 · Fungrim entry ↗

$\mathrm{BernoulliPolynomial}(n, 1)=(-1)^{n}\mathrm{BernoulliB}(n)$

$\mathrm{BernoulliPolynomial}(n, x+1)=\mathrm{BernoulliPolynomial}(n, x)+nx^{n-1}$

Holds when $n\in\N\land x\in\C$ . Symbols: BernoulliPolynomial — Bernoulli polynomial. Used by the Compute Engine for simplification. 8b4f7f · Fungrim entry ↗

$\mathrm{BernoulliPolynomial}(n, 0)=\mathrm{BernoulliB}(n)$

Holds when $n\in\N$ . Symbols: BernoulliB — Bernoulli number; BernoulliPolynomial — Bernoulli polynomial. Used by the Compute Engine for expansion. a1d2d7 · Fungrim entry ↗

$\mathrm{BernoulliB}(2n+3)=0$

Holds when $n\in\N$ . Symbols: BernoulliB — Bernoulli number. Used by the Compute Engine for simplification. a98234 · Fungrim entry ↗

$\mathrm{BernoulliPolynomial}(n, 1-x)=(-1)^{n}\mathrm{BernoulliPolynomial}(n, x)$

Holds when $n\in\N\land x\in\C$ . Symbols: BernoulliPolynomial — Bernoulli polynomial. Used by the Compute Engine for expansion. c2dcfa · Fungrim entry ↗

$x\mapsto\mathrm{BernoulliPolynomial}(n, x)^{\prime}(x)=n\mathrm{BernoulliPolynomial}(n-1, x)$

Holds when $n\in\N^*\land x\in\C$ . Symbols: BernoulliPolynomial — Bernoulli polynomial. Used by the Compute Engine for simplification. e89eb5 · Fungrim entry ↗

$\mathrm{BernoulliPolynomial}(n, -x)=(-1)^{n}(\mathrm{BernoulliPolynomial}(n, x)+nx^{n-1})$

Holds when $n\in\N\land x\in\C$ . Symbols: BernoulliPolynomial — Bernoulli polynomial. Used by the Compute Engine for simplification. f80439 · Fungrim entry ↗

Fibonacci numbers

$\mathrm{Fibonacci}(n)=\lfloor\frac{\varphi^{n}}{\sqrt{5}}+\frac{1}{2}\rfloor$

Holds when $n\in\N$ . Used by the Compute Engine for simplification. 050fdb · Fungrim entry ↗

$\mathrm{Fibonacci}(n)^2=\mathrm{Fibonacci}(n+1)\mathrm{Fibonacci}(n-1)-(-1)^{n}$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. 073466 · Fungrim entry ↗

$\begin{pmatrix}\mathrm{Fibonacci}(n+1)\\ \mathrm{Fibonacci}(n)\end{pmatrix}=\begin{pmatrix}1 & 1\\ 1 & 0\end{pmatrix}\begin{pmatrix}\mathrm{Fibonacci}(n)\\ \mathrm{Fibonacci}(n-1)\end{pmatrix}$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. 0e2425 · Fungrim entry ↗

$\mathrm{Fibonacci}(n+2)=\mathrm{Fibonacci}(n+1)+\mathrm{Fibonacci}(n)$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. 10165f · Fungrim entry ↗

$\mathrm{Fibonacci}(n)=(\exp(n\ln(\varphi))-\cos(\pi n)\exp(-n\ln(\varphi)))/\sqrt{5}$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. 12b336 · Fungrim entry ↗

$\mathrm{Fibonacci}(n)=\frac{n\mathrm{Hypergeometric2F_1}(\frac{1-n}{2}, \frac{2-n}{2}, \frac{3}{2}, 5)}{2^{n-1}}$

Holds when $n\in\Z$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. Reference: functions.wolfram.com 1c90fb · Fungrim entry ↗

$\mathrm{Fibonacci}(2n+1)=(2\mathrm{ChebyshevT}(2n+1, \frac{5^{1/2}}{2}))/\sqrt{5}$

Holds when $n\in\Z$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. 223ce1 · Fungrim entry ↗

$\mathrm{Fibonacci}(n)=\mathrm{Fibonacci}(n-1)+\mathrm{Fibonacci}(n-2)$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. 22dc6e · Fungrim entry ↗

$\mathrm{Fibonacci}(n)=(\varphi^{n}-(-\varphi)^{-n})/\sqrt{5}$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. 24107d · Fungrim entry ↗

$\mathrm{Fibonacci}(2n)=(\mathrm{Fibonacci}(n+1)+\mathrm{Fibonacci}(n-1))\mathrm{Fibonacci}(n)$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. 2ca869 · Fungrim entry ↗

$\mathrm{Fibonacci}(m)\mathrm{Fibonacci}(n+1)-\mathrm{Fibonacci}(m+1)\mathrm{Fibonacci}(n)=(-1)^{n}\mathrm{Fibonacci}(m-n)$

Holds when $n\in\Z\land m\in\Z$ . Used by the Compute Engine for simplification. 301081 · Fungrim entry ↗

$\mathrm{Fibonacci}(2n)=\mathrm{Fibonacci}(n+1)^2-\mathrm{Fibonacci}(n-1)^2$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. 35956b · Fungrim entry ↗

$\begin{pmatrix}\mathrm{Fibonacci}(n+m)\\ \mathrm{Fibonacci}((n+m)-1)\end{pmatrix}=\begin{pmatrix}1 & 1\\ 1 & 0\end{pmatrix}^{m}\begin{pmatrix}\mathrm{Fibonacci}(n)\\ \mathrm{Fibonacci}(n-1)\end{pmatrix}$

Holds when $n\in\Z\land m\in\Z$ . Used by the Compute Engine for expansion. 3a9c67 · Fungrim entry ↗

$\mathrm{Count}(\lbrace k, k\in\Z\in(n\mid\mathrm{Fibonacci}(k))\rbrace)=\mathrm{Count}(\Z)$

Holds when $n\in\Z\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 4ec333 · Fungrim entry ↗

$\mathrm{Fibonacci}(2n)=\mathrm{Fibonacci}(n+2)^2-\mathrm{Fibonacci}(n+1)^2-2\mathrm{Fibonacci}(n)^2$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. 5745bd · Fungrim entry ↗

$\mathrm{Fibonacci}(n)=\mathrm{Fibonacci}(m+1)\mathrm{Fibonacci}(n-m)+\mathrm{Fibonacci}(m)\mathrm{Fibonacci}(n-m-1)$

Holds when $m\in\Z\land n\in\Z$ . Used by the Compute Engine for simplification. 5cb57e · Fungrim entry ↗

$\mathrm{Fibonacci}(n)=\mathrm{Fibonacci}(n+2)-\mathrm{Fibonacci}(n+1)$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. 6d437c · Fungrim entry ↗

$\mathrm{Fibonacci}((m+n)-1)=\mathrm{Fibonacci}(m)\mathrm{Fibonacci}(n)+\mathrm{Fibonacci}(m-1)\mathrm{Fibonacci}(n-1)$

Holds when $m\in\Z\land n\in\Z$ . Used by the Compute Engine for simplification. 70878b · Fungrim entry ↗

$\gcd(\mathrm{Fibonacci}(n), \mathrm{Fibonacci}(n+1))=1$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. 7b0abf · Fungrim entry ↗

$\mathrm{Fibonacci}(n)=2\mathrm{Fibonacci}(n-2)+\mathrm{Fibonacci}(n-3)$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. 7ef2c7 · Fungrim entry ↗

$\begin{pmatrix}1 & 1\\ 1 & 0\end{pmatrix}^{n}=\begin{pmatrix}\mathrm{Fibonacci}(n+1) & \mathrm{Fibonacci}(n)\\ \mathrm{Fibonacci}(n) & \mathrm{Fibonacci}(n-1)\end{pmatrix}$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. 8a548e · Fungrim entry ↗

$\mathrm{Fibonacci}(n+i_{var})\mathrm{Fibonacci}(n+j)-\mathrm{Fibonacci}(n)\mathrm{Fibonacci}(n+i_{var}+j)=(-1)^{n}\mathrm{Fibonacci}(i_{var})\mathrm{Fibonacci}(j)$

Holds when $n\in\Z\land i_{var}\in\Z\land j\in\Z$ . Used by the Compute Engine for simplification. 8db61e · Fungrim entry ↗

$\mathrm{Fibonacci}(n)=\mathrm{Hypergeometric2F_1}(\frac{1-n}{2}, \frac{2-n}{2}, 1-n, -4)$

Holds when $n\in\N^*$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 90c290 · Fungrim entry ↗

$\mathrm{Fibonacci}(m+n)=\mathrm{Fibonacci}(m)\mathrm{Fibonacci}(n+1)+\mathrm{Fibonacci}(m-1)\mathrm{Fibonacci}(n)$

Holds when $m\in\Z\land n\in\Z$ . Used by the Compute Engine for simplification. a104b0 · Fungrim entry ↗

$\mathrm{Fibonacci}(n+1)=\mathrm{Fibonacci}(n)+\mathrm{Fibonacci}(n-1)$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. a8f2ac · Fungrim entry ↗

$\gcd(\mathrm{Fibonacci}(n), \mathrm{Fibonacci}(n+2))=1$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. aaa244 · Fungrim entry ↗

$\mathrm{Fibonacci}(2n)=\mathrm{ChebyshevU}(n-1, \frac{3}{2})$

Holds when $n\in\Z$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. aadf90 · Fungrim entry ↗

$\mathrm{Fibonacci}(n)^2=\mathrm{Fibonacci}(n+m)\mathrm{Fibonacci}(n-m)+(-1)^{n+m}\mathrm{Fibonacci}(m)^2$

Holds when $n\in\Z\land m\in\Z$ . Used by the Compute Engine for simplification. ab563e · Fungrim entry ↗

$\mathrm{Fibonacci}(n)=(\varphi^{n}-\frac{\cos(\pi n)}{\varphi^{n}})/\sqrt{5}$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. ad0d7a · Fungrim entry ↗

$\mathrm{Fibonacci}(n)=\imaginaryI^{n-1}\mathrm{ChebyshevU}(n-1, -(\frac{\imaginaryI}{2}))$

Holds when $n\in\Z$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. ae76a3 · Fungrim entry ↗

$\mathrm{Fibonacci}(n)=((1+\cos(\pi n))\sinh(n\ln(\varphi))+(1-\cos(\pi n))\cosh(n\ln(\varphi)))/\sqrt{5}$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. bceed4 · Fungrim entry ↗

$\mathrm{Fibonacci}(n)=(2(-\imaginaryI)^{n})/\sqrt{5}\sinh(n(\ln(\varphi)+\frac{\pi}{2}\imaginaryI))$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. c4d78a · Fungrim entry ↗

$\mathrm{Fibonacci}(n)=3\mathrm{Fibonacci}(n-3)+2\mathrm{Fibonacci}(n-4)$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. cbfe21 · Fungrim entry ↗

$\mathrm{Fibonacci}(-n)=(-1)^{n+1}\mathrm{Fibonacci}(n)$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. ce6dd0 · Fungrim entry ↗

$\mathrm{Fibonacci}(2n+1)=\mathrm{Fibonacci}(n+1)^2+\mathrm{Fibonacci}(n)^2$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. fc4fd1 · Fungrim entry ↗

$\mathrm{Fibonacci}(n)=(2\begin{cases}\sinh(n\ln(\varphi))&\mathrm{IsEven}(n)\\\cosh(n\ln(\varphi))&\mathrm{IsOdd}(n)\end{cases})/\sqrt{5}$

Holds when $n\in\Z$ . Used by the Compute Engine for simplification. fd732d · Fungrim entry ↗

Integer sequences

$\mathrm{Fibonacci}(n)=\mathrm{SloaneA}(\text{A000045}, n)$

Holds when $n\in\N$ . Symbols: SloaneA — Sequence X in Sloane's OEIS. Used by the Compute Engine for simplification. 373aa1 · Fungrim entry ↗

$\mathrm{PrimePi}(n)=\mathrm{SloaneA}(\text{A000720}, n)$

Holds when $n\in\N$ . Symbols: PrimePi — Prime counting function; SloaneA — Sequence X in Sloane's OEIS. Used by the Compute Engine for simplification. 4fa169 · Fungrim entry ↗

$\mathrm{BellNumber}(n)=\mathrm{SloaneA}(\text{A000110}, n)$

Holds when $n\in\N$ . Symbols: SloaneA — Sequence X in Sloane's OEIS. Used by the Compute Engine for simplification. 60dc3e · Fungrim entry ↗

$\mathrm{LandauG}(n)=\mathrm{SloaneA}(\text{A000793}, n)$

Holds when $n\in\N$ . Symbols: LandauG — Landau's function; SloaneA — Sequence X in Sloane's OEIS. Used by the Compute Engine for simplification. 6af603 · Fungrim entry ↗

$\mathrm{NPartition}(n)=\mathrm{SloaneA}(\text{A000041}, n)$

Holds when $n\in\N$ . Symbols: SloaneA — Sequence X in Sloane's OEIS. Used by the Compute Engine for simplification. 8eed2c · Fungrim entry ↗

$\mathrm{PrimeNumber}(n)=\mathrm{SloaneA}(\text{A000040}, n)$

Holds when $n\in\N^*$ . Symbols: PrimeNumber — nth prime number; SloaneA — Sequence X in Sloane's OEIS. Used by the Compute Engine for simplification. 9d0839 · Fungrim entry ↗

$\mathrm{BernoulliB}(n)=\frac{\mathrm{SloaneA}(\text{A027641}, n)}{\mathrm{SloaneA}(\text{A027642}, n)}$

Holds when $n\in\N$ . Symbols: BernoulliB — Bernoulli number; SloaneA — Sequence X in Sloane's OEIS. Used by the Compute Engine for simplification. b6111c · Fungrim entry ↗

$n!=\mathrm{SloaneA}(\text{A000142}, n)$

Holds when $n\in\N$ . Symbols: SloaneA — Sequence X in Sloane's OEIS. Used by the Compute Engine for simplification. d12aa0 · Fungrim entry ↗

Partition function

$\mathrm{NPartition}(4)=\mathrm{Count}(\lbrace\bigl\lbrack4\bigr\rbrack, \bigl\lbrack3, 1\bigr\rbrack, \bigl\lbrack2, 2\bigr\rbrack, \bigl\lbrack2, 1, 1\bigr\rbrack, \bigl\lbrack1, 1, 1, 1\bigr\rbrack\rbrace)=5$

Used by the Compute Engine for simplification. 6018a4 · Fungrim entry ↗

$\mathrm{NPartition}(3)=\mathrm{Count}(\lbrace\bigl\lbrack3\bigr\rbrack, \bigl\lbrack2, 1\bigr\rbrack, \bigl\lbrack1, 1, 1\bigr\rbrack\rbrace)=3$

Used by the Compute Engine for simplification. 7ef291 · Fungrim entry ↗

$\mathrm{NPartition}(2)=\mathrm{Count}(\lbrace\bigl\lbrack2\bigr\rbrack, \bigl\lbrack1, 1\bigr\rbrack\rbrace)=2$

Used by the Compute Engine for simplification. b2583f · Fungrim entry ↗

$\mathrm{NPartition}(-n)=0$

Holds when $n\in\N^*$ . Used by the Compute Engine for simplification. cd3013 · Fungrim entry ↗

$\mathrm{NPartition}(0)=\mathrm{Count}(\lbrace\bigl\lbrack \bigr\rbrack\rbrace)=1$

Used by the Compute Engine for simplification. cebe1b · Fungrim entry ↗

$\mathrm{NPartition}(1)=\mathrm{Count}(\lbrace\bigl\lbrack1\bigr\rbrack\rbrace)=1$

Used by the Compute Engine for simplification. e84642 · Fungrim entry ↗

Stirling numbers

$\mathrm{StirlingS_1}(n, k)=(-1)^{n+k}\mathrm{StirlingCycle}(n, k)$

Holds when $n\in\N\land k\in\N$ . Symbols: StirlingCycle — Unsigned Stirling number of the first kind; StirlingS1 — Signed Stirling number of the first kind. Used by the Compute Engine for simplification. 071a94 · Fungrim entry ↗

$\mathrm{StirlingS_1}(n+1, k)=\mathrm{StirlingS_1}(n, k-1)-n\mathrm{StirlingS_1}(n, k)$

Holds when $n\in\N\land k\in\N^*$ . Symbols: StirlingS1 — Signed Stirling number of the first kind. Used by the Compute Engine for simplification. 18ec99 · Fungrim entry ↗

$\mathrm{Stirling}(n+1, k)=k\mathrm{Stirling}(n, k)+\mathrm{Stirling}(n, k-1)$

Holds when $n\in\N\land k\in\N^*$ . Used by the Compute Engine for simplification. 9fbe4f · Fungrim entry ↗

$\mathrm{StirlingCycle}(n+1, k)=n\mathrm{StirlingCycle}(n, k)+\mathrm{StirlingCycle}(n, k-1)$

Holds when $n\in\N\land k\in\N^*$ . Symbols: StirlingCycle — Unsigned Stirling number of the first kind. Used by the Compute Engine for simplification. f0d72c · Fungrim entry ↗

Contents​

Bell numbers​

Bernoulli numbers and polynomials​

Fibonacci numbers​

Integer sequences​

Partition function​

Stirling numbers​

Contents