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Let $a(n)$ be A309099 i.e. the number of partitions of $n$ avoiding the partition $(4,3,1)$. We say a partition $\alpha$ contains $\mu$ provided that one can delete rows and columns from (the Ferrer …

Let $a(n,m,k)$ be an integer sequence such that $$a(n,m,k)=\sum\limits_{i=0}^{n}{n\brace i}(m-k)^{n-i}\prod\limits_{j=0}^{i-1}(kj+1)$$ Here ${n\brace k}$ is the Stirling number of the second kind. The …

Let $p(n)$ be A000041 i.e. number of partitions of $n$ (the partition numbers). Let $$ \ell(n)=\left\lfloor\log_2 n\right\rfloor $$ Let $a(n)$ be A161511 i.e. number of $1\cdots0$ pairs in the bina …

Let $a(n)$ be A110501 (i.e., unsigned Genocchi numbers (of first kind) of even index). Here $$ a(n) = \sum\limits_{i=1}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n}{2i}a(n-i)(-1)^{i-1}, \\ a(1) = 1 …

Let $q(n)$ be A007814, i.e., the number of trailing zeros in the binary representation of $n$. Here $$q(2n+1)=0, q(2n)=q(n)+1$$ Let $a(n)$ be A329369. Here $$a(2n+1)=a(n), a(2n)=a(n)+a(n-2^{q(n)})+a(2 …

Let $a_1(n)$ be A000108, i.e. Catalan numbers. Here $$ a_1(n)=\frac{1}{n+1}\binom{2n}{n} $$ Let $a_2(n)$ be A059715, i.e. number of multi-directed animals on the triangular lattice. From OEIS page we …

Let $s(n,k)$ be a (signed) Stirling number of the first kind. Let $n \brace k$ be a Stirling number of the second kind. Let $a(n)$ be A014307. Here $$ A(x) = \sum\limits_{k=0}^{\infty} \frac{a(k) …

Working on suitable closed form for A329369, I discovered very useful coefficients, which have the following recurrence relation: $$T(0,1)=T(0,2)=1$$ $$T(n,1)=1, n>0$$ $$T(0,k)=0, k>2$$ $$T(2n+1,k)=kT …

Let $q(n)$ be A007814, i.e., number of trailing zeros in the binary representation of $n$. Here $$q(2n+1)=0, q(2n)=q(n)+1$$ Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary expan …

Let $a(n)$ be A002720, i.e., number of partial permutations of an $n$-set; number of $n \times n$ binary matrices with at most one $1$ in each row and column. $$a(n)=\sum\limits_{k=0}^{n} k!\binom{n}{ …

Let $q(n)$ be A007814, i.e., number of trailing zeros in the binary representation of $n$. Here $$q(2n+1)=0, q(2n)=q(n)+1$$ Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary expan …

Let $a(n)$ be A143017 i.e. number of $\{2-1-3, 2'^e-31\}$-avoiding permutations of size $n$ (see definition in the Elizalde paper). Here $$ a(n) = \frac{1}{n}\sum\limits_{k=0}^{\left\lfloor\frac{n} …

Let $a(n)$ be A089039 (i.e., number of circular permutations of $2n$ letters that are free of jealousy). Here $$ a(n) = \sum\limits_{k=1}^{\left\lfloor\frac{n}{2}\right\rfloor}\frac{n!(n-k-1)!^2}{(k …

Let $s(n,k)$ be a (signed) Stirling number of the first kind. Let ${n \brace k}$ be a Stirling number of the second kind. Let $$ f(n,m,i) = (-1)^{m-i+1}\sum\limits_{j=i}^{m+1}j^n s(j,i) {m+1 \bra …

Given $n$ balls, all of which are initially in the first of $n$ numbered boxes, $a(n)$ is the number of steps required to get one ball in each box when a step consists of moving to the next box every  …