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Let $$ \ell(n) = \left\lfloor\log_2 n\right\rfloor. $$ Let $T(n,k)$ be an integer coefficients with row length $f(n)$ (number of zeros in the binary expansion of $n$ plus $2$ for $n>0$ with $f(0)=1$) …

Let $\operatorname{wt}(n)$ be A000120 (i.e., number of $1$'s in binary expansion of $n$). Let $a(n,m)$ be the family of integer sequences such that $$ a(n,m) = \sum\limits_{k=0}^{n} [\operatorname{wt …

Let $$ \ell(n) = \left\lfloor\log_2 n\right\rfloor, \\ \ell(0) = -1 $$ Let $$ f(n) = \ell(n) - \ell(n-2^{\ell(n)}) - 1 $$ Here $f(n)$ is A290255. Let $A(n,k)$ be a square array such that $$ A(n, …

Let $f(n)$ be an arbitrary function. Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). Here $$ \operatorname{wt}(2n+1) = \operatorname{wt}(n) + 1, \\ \opera …

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) …

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 …

Let $s(n,k)$ be a (signed) Stirling number of the first kind. Let $R(n,x)$ be the family of polynomials such that $$ R(2n+1,x) = xR(n,x), \\ R(2n,x) = x(R(n,x+1) - R(n, x)), \\ R(0, x) = x $$ Let …

Let $a(n)$ be A329369 (i.e., number of permutations of $\{1,2,\dotsc,m\}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \leqslant …

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 $a_1(n)$ 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 $a(n)$ 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 $a(n)$ 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 $p(n)$ 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 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 $q(n)$ 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 …