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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 $T(n,k)$ be A129179 (i.e., triangle read by rows: $T(n, k)$ is the number of Schroeder paths of semilength $n$ such that the area between the $x$-axis and the path is $k$ ($n \geqslant 0, 0 \leqs …

Let $a(n)$ be A088352. Here $a(n)$ is an integer sequence with generating function $A(x)$ such that $$ A(x) = \cfrac{1}{1-x-\cfrac{x^2}{1-x^3-\cfrac{x^4}{1-x^5-\cfrac{x^6}{1-x^7-\cfrac{x^8}{\ddots}}} …

Let $a(n)$ be A347205. Here $$ a(2^m(2k+1)) = \sum\limits_{j=0}^{m}a(2^j k), \\ a(0) = 1. $$ Let $\nu_2(n)$ be A007814 (i.e., number of trailing zeros in the binary expansion of $n$). Here $$ \nu_2(2 …

Let $f(n)$ be an arbitrary function with integer values. Let $a(n)$ be an integer sequence such that $$ \frac{1}{1-x}=\sum\limits_{n=0}^{\infty}a(n)x^n\prod\limits_{k=1}^{n+1}(1-f(k)x) $$ Start with …

Let $a(n)$ be A162326. Here $$ a(n) = \frac{1}{n}(2(5n-7)a(n-1) - 9(n-2)a(n-2)), \\ a(0) = a(1) = 1. $$ Also ordinary generating function is $$ \frac{5 - \sqrt{\frac{1-9x}{1-x}}}{4}. $$ Let $b(n)$ be …

Let ${n \brace k}$ be a Stirling number of the second kind. Let $A_n(x)$ be an Eulerian polynomial. Here $$ A_n(x) = \sum_{i=0}^{n}i!{n \brace i}(x-1)^{n-i}. $$ Let $a_1(n,p,q)$ be the family of inte …

Let $a(n,p,q)$ be the family of integer sequences such that ordinary generating functions for it are $\frac{1}{G_1(0,x)}$ where $G_1(0,x)$ are continued fractions such that $$ G_1(j,x)=1-\cfrac{(qj+1 …

Let $G_n$ be A036968 (i.e., Genocchi numbers). Here $$ \frac{2t}{1+e^t}=\sum\limits_{n=0}^{\infty}G_n\frac{t^n}{n!}. $$ Also $$ t\tan\left(\frac{t}{2}\right)=\sum\limits_{n=1}^{\infty}(-1)^n G_{2n}\f …

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 $a(n,p,q)$ be the family of integer sequences such that exponential generating functions for it satisfy $$ A_1(x)=\exp\left(x + p\int\int (A_1(x))^q \, dx \, dx\right). $$ Let $b(n,p,q)$ be the …

Let $$ T(n,k,p,q,r,s) = (q(k-1)+1)T(n-1,k,p,q,r,s) + s(n+r(k-1)+p-2)T(n-1,k-1,p,q,r,s), \\ T(n,1,p,q,r,s) = 1, \\ T(n,0,p,q,r,s) = T(0,k,p,q,r,s) = 0 $$ Let $$ \ell(n) = \left\lfloor\log_2 n\right …

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, …