# Combinatorial and integer sequences Part of the [Fungrim Identities](/compute-engine/reference/fungrim/) reference — **65 identities** for combinatorial and integer sequences. :::info[Generated reference] This page is generated from the compiled Fungrim artifact by `scripts/fungrim/gen-reference-doc.ts` (upstream snapshot `3a299164c683`, translator `grim2mathjson 0.1.0`). Do not edit it by hand. The corpus is MIT-licensed; see `data/fungrim/LICENSE`. ::: ## Contents - [Bell numbers](#bell-numbers) (1) - [Bernoulli numbers and polynomials](#bernoulli-numbers-and-polynomials) (9) - [Fibonacci numbers](#fibonacci-numbers) (37) - [Integer sequences](#integer-sequences) (8) - [Partition function](#partition-function) (6) - [Stirling numbers](#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](https://arxiv.org/abs/0708.3301) [`f4e249` · Fungrim entry ↗](https://fungrim.org/entry/f4e249) --- ## Bernoulli numbers and polynomials $$\mathrm{BernoulliPolynomial}(n, \frac{1}{2})=(2^{1-n}-1)\mathrm{BernoulliB}(n)$$ **Holds when** $n\in\N$. **Symbols:** **BernoulliPolynomial** — Bernoulli polynomial. Used by the Compute Engine for simplification. [`03ee0b` · Fungrim entry ↗](https://fungrim.org/entry/03ee0b) --- $$\mathrm{BernoulliB}(2n)=\frac{2\Zeta(2n)(2n)!\times(-1)^{n+1}}{(2\pi)^{2n}}$$ **Holds when** $n\in\N^*$. Used by the Compute Engine for simplification. [`14ecc4` · Fungrim entry ↗](https://fungrim.org/entry/14ecc4) --- $$\mathrm{BernoulliPolynomial}(n, 1)=(-1)^{n}\mathrm{BernoulliB}(n)$$ **Holds when** $n\in\N$. **Symbols:** **BernoulliPolynomial** — Bernoulli polynomial. Used by the Compute Engine for simplification. [`829185` · Fungrim entry ↗](https://fungrim.org/entry/829185) --- $$\mathrm{BernoulliPolynomial}(n, x+1)=\mathrm{BernoulliPolynomial}(n, x)+nx^{n-1}$$ **Holds when** $n\in\N\land x\in\C$. **Symbols:** **BernoulliPolynomial** — Bernoulli polynomial. Used by the Compute Engine for simplification. [`8b4f7f` · Fungrim entry ↗](https://fungrim.org/entry/8b4f7f) --- $$\mathrm{BernoulliPolynomial}(n, 0)=\mathrm{BernoulliB}(n)$$ **Holds when** $n\in\N$. **Symbols:** **BernoulliPolynomial** — Bernoulli polynomial. Used by the Compute Engine for expansion. [`a1d2d7` · Fungrim entry ↗](https://fungrim.org/entry/a1d2d7) --- $$\mathrm{BernoulliB}(2n+3)=0$$ **Holds when** $n\in\N$. Used by the Compute Engine for simplification. [`a98234` · Fungrim entry ↗](https://fungrim.org/entry/a98234) --- $$\mathrm{BernoulliPolynomial}(n, 1-x)=(-1)^{n}\mathrm{BernoulliPolynomial}(n, x)$$ **Holds when** $n\in\N\land x\in\C$. **Symbols:** **BernoulliPolynomial** — Bernoulli polynomial. Used by the Compute Engine for expansion. [`c2dcfa` · Fungrim entry ↗](https://fungrim.org/entry/c2dcfa) --- $$x\mapsto\mathrm{BernoulliPolynomial}(n, x)^{\prime}(x)=n\mathrm{BernoulliPolynomial}(n-1, x)$$ **Holds when** $n\in\N^*\land x\in\C$. **Symbols:** **BernoulliPolynomial** — Bernoulli polynomial. Used by the Compute Engine for simplification. [`e89eb5` · Fungrim entry ↗](https://fungrim.org/entry/e89eb5) --- $$\mathrm{BernoulliPolynomial}(n, -x)=(-1)^{n}(\mathrm{BernoulliPolynomial}(n, x)+nx^{n-1})$$ **Holds when** $n\in\N\land x\in\C$. **Symbols:** **BernoulliPolynomial** — Bernoulli polynomial. Used by the Compute Engine for simplification. [`f80439` · Fungrim entry ↗](https://fungrim.org/entry/f80439) --- ## 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 ↗](https://fungrim.org/entry/050fdb) --- $$\mathrm{Fibonacci}(n)^2=\mathrm{Fibonacci}(n+1)\mathrm{Fibonacci}(n-1)-(-1)^{n}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`073466` · Fungrim entry ↗](https://fungrim.org/entry/073466) --- $$\begin{pmatrix}\mathrm{Fibonacci}(n+1)\\ \mathrm{Fibonacci}(n)\end{pmatrix}=\begin{pmatrix}1 & 1\\ 1 & 0\end{pmatrix}\begin{pmatrix}\mathrm{Fibonacci}(n)\\ \mathrm{Fibonacci}(n-1)\end{pmatrix}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`0e2425` · Fungrim entry ↗](https://fungrim.org/entry/0e2425) --- $$\mathrm{Fibonacci}(n+2)=\mathrm{Fibonacci}(n+1)+\mathrm{Fibonacci}(n)$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`10165f` · Fungrim entry ↗](https://fungrim.org/entry/10165f) --- $$\mathrm{Fibonacci}(n)=(\exp(n\ln(\varphi))-\cos(\pi n)\exp(-n\ln(\varphi)))/\sqrt{5}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`12b336` · Fungrim entry ↗](https://fungrim.org/entry/12b336) --- $$\mathrm{Fibonacci}(n)=\frac{n\mathrm{Hypergeometric2F_1}(\frac{1-n}{2}, \frac{2-n}{2}, \frac{3}{2}, 5)}{2^{n-1}}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. **Reference:** [functions.wolfram.com](http://functions.wolfram.com/IntegerFunctions/Fibonacci/26/01/01/0007/) [`1c90fb` · Fungrim entry ↗](https://fungrim.org/entry/1c90fb) --- $$\mathrm{Fibonacci}(2n+1)=(2\mathrm{ChebyshevT}(2n+1, \frac{5^{1/2}}{2}))/\sqrt{5}$$ **Holds when** $n\in\Z$. **Symbols:** **ChebyshevT** — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. [`223ce1` · Fungrim entry ↗](https://fungrim.org/entry/223ce1) --- $$\mathrm{Fibonacci}(n)=\mathrm{Fibonacci}(n-1)+\mathrm{Fibonacci}(n-2)$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`22dc6e` · Fungrim entry ↗](https://fungrim.org/entry/22dc6e) --- $$\mathrm{Fibonacci}(n)=(\varphi^{n}-(-\varphi)^{-n})/\sqrt{5}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`24107d` · Fungrim entry ↗](https://fungrim.org/entry/24107d) --- $$\mathrm{Fibonacci}(2n)=(\mathrm{Fibonacci}(n+1)+\mathrm{Fibonacci}(n-1))\mathrm{Fibonacci}(n)$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`2ca869` · Fungrim entry ↗](https://fungrim.org/entry/2ca869) --- $$\mathrm{Fibonacci}(m)\mathrm{Fibonacci}(n+1)-\mathrm{Fibonacci}(m+1)\mathrm{Fibonacci}(n)=(-1)^{n}\mathrm{Fibonacci}(m-n)$$ **Holds when** $n\in\Z\land m\in\Z$. Used by the Compute Engine for simplification. [`301081` · Fungrim entry ↗](https://fungrim.org/entry/301081) --- $$\mathrm{Fibonacci}(2n)=\mathrm{Fibonacci}(n+1)^2-\mathrm{Fibonacci}(n-1)^2$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`35956b` · Fungrim entry ↗](https://fungrim.org/entry/35956b) --- $$\begin{pmatrix}\mathrm{Fibonacci}(n+m)\\ \mathrm{Fibonacci}((n+m)-1)\end{pmatrix}=\begin{pmatrix}1 & 1\\ 1 & 0\end{pmatrix}^{m}\begin{pmatrix}\mathrm{Fibonacci}(n)\\ \mathrm{Fibonacci}(n-1)\end{pmatrix}$$ **Holds when** $n\in\Z\land m\in\Z$. Used by the Compute Engine for expansion. [`3a9c67` · Fungrim entry ↗](https://fungrim.org/entry/3a9c67) --- $$\mathrm{Count}(\mathrm{Filter}(\Z, k\mapsto n\mid\mathrm{Fibonacci}(k)))=\mathrm{Count}(\Z)$$ **Holds when** $n\in\Z\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`4ec333` · Fungrim entry ↗](https://fungrim.org/entry/4ec333) --- $$\mathrm{Fibonacci}(2n)=\mathrm{Fibonacci}(n+2)^2-\mathrm{Fibonacci}(n+1)^2-2\mathrm{Fibonacci}(n)^2$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`5745bd` · Fungrim entry ↗](https://fungrim.org/entry/5745bd) --- $$\mathrm{Fibonacci}(n)=\mathrm{Fibonacci}(m+1)\mathrm{Fibonacci}(n-m)+\mathrm{Fibonacci}(m)\mathrm{Fibonacci}(n-m-1)$$ **Holds when** $m\in\Z\land n\in\Z$. Used by the Compute Engine for simplification. [`5cb57e` · Fungrim entry ↗](https://fungrim.org/entry/5cb57e) --- $$\mathrm{Fibonacci}(n)=\mathrm{Fibonacci}(n+2)-\mathrm{Fibonacci}(n+1)$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`6d437c` · Fungrim entry ↗](https://fungrim.org/entry/6d437c) --- $$\mathrm{Fibonacci}((m+n)-1)=\mathrm{Fibonacci}(m)\mathrm{Fibonacci}(n)+\mathrm{Fibonacci}(m-1)\mathrm{Fibonacci}(n-1)$$ **Holds when** $m\in\Z\land n\in\Z$. Used by the Compute Engine for simplification. [`70878b` · Fungrim entry ↗](https://fungrim.org/entry/70878b) --- $$\gcd(\mathrm{Fibonacci}(n), \mathrm{Fibonacci}(n+1))=1$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`7b0abf` · Fungrim entry ↗](https://fungrim.org/entry/7b0abf) --- $$\mathrm{Fibonacci}(n)=2\mathrm{Fibonacci}(n-2)+\mathrm{Fibonacci}(n-3)$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`7ef2c7` · Fungrim entry ↗](https://fungrim.org/entry/7ef2c7) --- $$\begin{pmatrix}1 & 1\\ 1 & 0\end{pmatrix}^{n}=\begin{pmatrix}\mathrm{Fibonacci}(n+1) & \mathrm{Fibonacci}(n)\\ \mathrm{Fibonacci}(n) & \mathrm{Fibonacci}(n-1)\end{pmatrix}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`8a548e` · Fungrim entry ↗](https://fungrim.org/entry/8a548e) --- $$\mathrm{Fibonacci}(n+i_{var})\mathrm{Fibonacci}(n+j)-\mathrm{Fibonacci}(n)\mathrm{Fibonacci}(n+i_{var}+j)=(-1)^{n}\mathrm{Fibonacci}(i_{var})\mathrm{Fibonacci}(j)$$ **Holds when** $n\in\Z\land i_{var}\in\Z\land j\in\Z$. Used by the Compute Engine for simplification. [`8db61e` · Fungrim entry ↗](https://fungrim.org/entry/8db61e) --- $$\mathrm{Fibonacci}(n)=\mathrm{Hypergeometric2F_1}(\frac{1-n}{2}, \frac{2-n}{2}, 1-n, -4)$$ **Holds when** $n\in\N^*$. Used by the Compute Engine for simplification. [`90c290` · Fungrim entry ↗](https://fungrim.org/entry/90c290) --- $$\mathrm{Map}(\mathrm{Filter}(\N, n\mapsto\mathrm{Fibonacci}(n)^{1/2}\in\Z), n\mapsto\mathrm{Fibonacci}(n))=\lbrace\mathrm{Fibonacci}(0), \mathrm{Fibonacci}(1), \mathrm{Fibonacci}(2), \mathrm{Fibonacci}(12)\rbrace=\lbrace0, 1, 144\rbrace$$ Used by the Compute Engine for simplification. [`9d26d2` · Fungrim entry ↗](https://fungrim.org/entry/9d26d2) --- $$\mathrm{Fibonacci}(m+n)=\mathrm{Fibonacci}(m)\mathrm{Fibonacci}(n+1)+\mathrm{Fibonacci}(m-1)\mathrm{Fibonacci}(n)$$ **Holds when** $m\in\Z\land n\in\Z$. Used by the Compute Engine for simplification. [`a104b0` · Fungrim entry ↗](https://fungrim.org/entry/a104b0) --- $$\mathrm{Fibonacci}(n+1)=\mathrm{Fibonacci}(n)+\mathrm{Fibonacci}(n-1)$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`a8f2ac` · Fungrim entry ↗](https://fungrim.org/entry/a8f2ac) --- $$\gcd(\mathrm{Fibonacci}(n), \mathrm{Fibonacci}(n+2))=1$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`aaa244` · Fungrim entry ↗](https://fungrim.org/entry/aaa244) --- $$\mathrm{Fibonacci}(2n)=\mathrm{ChebyshevU}(n-1, \frac{3}{2})$$ **Holds when** $n\in\Z$. **Symbols:** **ChebyshevU** — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. [`aadf90` · Fungrim entry ↗](https://fungrim.org/entry/aadf90) --- $$\mathrm{Fibonacci}(n)^2=\mathrm{Fibonacci}(n+m)\mathrm{Fibonacci}(n-m)+(-1)^{n+m}\mathrm{Fibonacci}(m)^2$$ **Holds when** $n\in\Z\land m\in\Z$. Used by the Compute Engine for simplification. [`ab563e` · Fungrim entry ↗](https://fungrim.org/entry/ab563e) --- $$\mathrm{Fibonacci}(n)=(\varphi^{n}-\frac{\cos(\pi n)}{\varphi^{n}})/\sqrt{5}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`ad0d7a` · Fungrim entry ↗](https://fungrim.org/entry/ad0d7a) --- $$\mathrm{Fibonacci}(n)=\imaginaryI^{n-1}\mathrm{ChebyshevU}(n-1, -(\frac{\imaginaryI}{2}))$$ **Holds when** $n\in\Z$. **Symbols:** **ChebyshevU** — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. [`ae76a3` · Fungrim entry ↗](https://fungrim.org/entry/ae76a3) --- $$\mathrm{Fibonacci}(n)=((1+\cos(\pi n))\sinh(n\ln(\varphi))+(1-\cos(\pi n))\cosh(n\ln(\varphi)))/\sqrt{5}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`bceed4` · Fungrim entry ↗](https://fungrim.org/entry/bceed4) --- $$\mathrm{Fibonacci}(n)=(2(-\imaginaryI)^{n})/\sqrt{5}\sinh(n(\ln(\varphi)+\frac{\pi}{2}\imaginaryI))$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`c4d78a` · Fungrim entry ↗](https://fungrim.org/entry/c4d78a) --- $$\mathrm{Fibonacci}(n)=3\mathrm{Fibonacci}(n-3)+2\mathrm{Fibonacci}(n-4)$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`cbfe21` · Fungrim entry ↗](https://fungrim.org/entry/cbfe21) --- $$\mathrm{Fibonacci}(-n)=(-1)^{n+1}\mathrm{Fibonacci}(n)$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`ce6dd0` · Fungrim entry ↗](https://fungrim.org/entry/ce6dd0) --- $$\mathrm{Fibonacci}(2n+1)=\mathrm{Fibonacci}(n+1)^2+\mathrm{Fibonacci}(n)^2$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`fc4fd1` · Fungrim entry ↗](https://fungrim.org/entry/fc4fd1) --- $$\mathrm{Fibonacci}(n)=(2\begin{cases}\sinh(n\ln(\varphi))&\mathrm{IsEven}(n)\\\cosh(n\ln(\varphi))&\mathrm{IsOdd}(n)\end{cases})/\sqrt{5}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`fd732d` · Fungrim entry ↗](https://fungrim.org/entry/fd732d) --- ## 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 ↗](https://fungrim.org/entry/373aa1) --- $$\mathrm{PrimePi}(n)=\mathrm{SloaneA}(\text{A000720}, n)$$ **Holds when** $n\in\N$. **Symbols:** **SloaneA** — Sequence X in Sloane's OEIS. Used by the Compute Engine for simplification. [`4fa169` · Fungrim entry ↗](https://fungrim.org/entry/4fa169) --- $$\mathrm{BellNumber}(n)=\mathrm{SloaneA}(\text{A000110}, n)$$ **Holds when** $n\in\N$. **Symbols:** **SloaneA** — Sequence X in Sloane's OEIS. Used by the Compute Engine for simplification. [`60dc3e` · Fungrim entry ↗](https://fungrim.org/entry/60dc3e) --- $$\mathrm{LandauG}(n)=\mathrm{SloaneA}(\text{A000793}, n)$$ **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 ↗](https://fungrim.org/entry/6af603) --- $$\mathrm{NPartition}(n)=\mathrm{SloaneA}(\text{A000041}, n)$$ **Holds when** $n\in\N$. **Symbols:** **SloaneA** — Sequence X in Sloane's OEIS. Used by the Compute Engine for simplification. [`8eed2c` · Fungrim entry ↗](https://fungrim.org/entry/8eed2c) --- $$\mathrm{PrimeNumber}(n)=\mathrm{SloaneA}(\text{A000040}, n)$$ **Holds when** $n\in\N^*$. **Symbols:** **SloaneA** — Sequence X in Sloane's OEIS. Used by the Compute Engine for simplification. [`9d0839` · Fungrim entry ↗](https://fungrim.org/entry/9d0839) --- $$\mathrm{BernoulliB}(n)=\frac{\mathrm{SloaneA}(\text{A027641}, n)}{\mathrm{SloaneA}(\text{A027642}, n)}$$ **Holds when** $n\in\N$. **Symbols:** **SloaneA** — Sequence X in Sloane's OEIS. Used by the Compute Engine for simplification. [`b6111c` · Fungrim entry ↗](https://fungrim.org/entry/b6111c) --- $$n!=\mathrm{SloaneA}(\text{A000142}, n)$$ **Holds when** $n\in\N$. **Symbols:** **SloaneA** — Sequence X in Sloane's OEIS. Used by the Compute Engine for simplification. [`d12aa0` · Fungrim entry ↗](https://fungrim.org/entry/d12aa0) --- ## 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 ↗](https://fungrim.org/entry/6018a4) --- $$\mathrm{NPartition}(3)=\mathrm{Count}(\lbrace\bigl\lbrack3\bigr\rbrack, \bigl\lbrack2, 1\bigr\rbrack, \bigl\lbrack1, 1, 1\bigr\rbrack\rbrace)=3$$ Used by the Compute Engine for simplification. [`7ef291` · Fungrim entry ↗](https://fungrim.org/entry/7ef291) --- $$\mathrm{NPartition}(2)=\mathrm{Count}(\lbrace\bigl\lbrack2\bigr\rbrack, \bigl\lbrack1, 1\bigr\rbrack\rbrace)=2$$ Used by the Compute Engine for simplification. [`b2583f` · Fungrim entry ↗](https://fungrim.org/entry/b2583f) --- $$\mathrm{NPartition}(-n)=0$$ **Holds when** $n\in\N^*$. Used by the Compute Engine for simplification. [`cd3013` · Fungrim entry ↗](https://fungrim.org/entry/cd3013) --- $$\mathrm{NPartition}(0)=\mathrm{Count}(\lbrace\bigl\lbrack \bigr\rbrack\rbrace)=1$$ Used by the Compute Engine for simplification. [`cebe1b` · Fungrim entry ↗](https://fungrim.org/entry/cebe1b) --- $$\mathrm{NPartition}(1)=\mathrm{Count}(\lbrace\bigl\lbrack1\bigr\rbrack\rbrace)=1$$ Used by the Compute Engine for simplification. [`e84642` · Fungrim entry ↗](https://fungrim.org/entry/e84642) --- ## 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 ↗](https://fungrim.org/entry/071a94) --- $$\mathrm{StirlingS_1}(n+1, k)=\mathrm{StirlingS_1}(n, k-1)-n\mathrm{StirlingS_1}(n, k)$$ **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 ↗](https://fungrim.org/entry/18ec99) --- $$\mathrm{Stirling}(n+1, k)=k\mathrm{Stirling}(n, k)+\mathrm{Stirling}(n, k-1)$$ **Holds when** $n\in\N\land k\in\N^*$. Used by the Compute Engine for simplification. [`9fbe4f` · Fungrim entry ↗](https://fungrim.org/entry/9fbe4f) --- $$\mathrm{StirlingCycle}(n+1, k)=n\mathrm{StirlingCycle}(n, k)+\mathrm{StirlingCycle}(n, k-1)$$ **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 ↗](https://fungrim.org/entry/f0d72c) ---