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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 $f(0)=1$) …

It is known that Striling numbers of the second kind satisfy the relation $$ \sum\limits_{k=0}^{n}{n \brace k}(x)_k = x^n. $$ where $(x)_n$ is the falling factorials such that $$ (x)_n = x(x-1)(x-2)\d …

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 $a(n)$ be A302117. Here $$ a(n) = 4(n-1)a(n-1) - \frac{1}{3}\prod\limits_{k=0}^{n-1}(2k-3), \\ a(0) = 0. $$ Let $$ T(n,k) = \sum\limits_{i=0}^{k} \binom{n}{i}. $$ Let $P_n(z)$ be the family of po …

Let $a(n)$ be A108442. Here generating function is $\frac{1}{1-zA(z)}$ where $$ A(z) = 1 + z(A(z))^2 + z(A(z))^3. $$ Also $$ a(n) = \sum\limits_{k=1}^{n}\frac{k}{2n-k}\sum\limits_{i=0}^{n-k} \binom{2 …

Let $a(n)$ be A208832. Here $$ \frac{1}{1-x} = \sum\limits_{n=0}^{\infty}a(n)x^n\prod\limits_{k=1}^{n}\frac{1-kx}{1+kx}. $$ Start with vector $\nu$ of fixed length $m$ with elements $\nu_i = 1$ (that …

Let $a(n)$ be A217061. Here $$ a(n) = \sum\limits_{m=1}^{n}\frac{1}{(m-1)!}\sum\limits_{k=0}^{n-m}(n+k-1)!\sum\limits_{j=0}^{k}\frac{1}{(k-j)!}\sum\limits_{\ell=0}^{j}\frac{2^{\ell-j}(-1)^{\ell+j}s(n …

Let $a(n)$ be A006351 (i.e., number of series-parallel networks with n labeled edges. Also called yoke-chains by Cayley and MacMahon). Here exponential generating function is $A(x)$ such that $B(x) = …

Let $a(n)$ be A338193. Here generating function is $A(x)$ such that $$ A(x) = 1 + \int\frac{\left(\frac{x}{A(x)}\right)'}{\left(\frac{x}{(A(x))^2}\right)'} \, dx. $$ Let $$ R(n, q) = \begin{cases} 1 …

Let $a(n)$ be A033264 (i.e., number of blocks of $\{1,0\}$ in the binary expansion of $n$). Here $$ a(4n) = a(4n+1) = a(2n), \\ a(4n+2) = a(n)+1, \\ a(4n+3) = a(n), \\ a(0) = 0. $$ Let $$ \ell(n) = \ …

Let $a(n)$ be A057326 (i.e., first member of a prime triple in a $2p-1$ progression). Let $b(n) = B$ after $n-1$ iterations where we start with $A=n, B=1$ and for $i$ from $1$ to $n-1$ simultaneously …

Let $a(n)$ be A069999 (i.e., number of possible dimensions for commutators of $n \times n$ matrices; it is independent of the field). OEIS states that no generating function is known. Let $P(n,k)$ be …

Let $a(n)$ be A127782 (i.e., an integer sequence with generating function $A(x)$ such that $A(x)=1+xA(x+x^2)$). Here $$ a(n) = \sum\limits_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-k-1} …

Let $C_n$ be A000108 (i.e., Catalan numbers). Here generating function is $C(x)$ such that $$ C(x) = \frac{1-\sqrt{1-4x}}{2x}. $$ Let $a(n)$ be an integer sequence with generating function $A(x)$ suc …

Let $a(n)$ be A258173 i.e. sum over all Dyck paths of semilength $n$ of products over all peaks $p$ of $y_p$, where $y_p$ is the $y$-coordinate of peak $p$. A Dyck path of semilength $n$ is a $(x,y)$ …