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Let $p$ and $q$ be integers. Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$. Then …

Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$. Let $a(n)$ be an arbitrary integer sequence such that $a(0)=1$. Let $b(n)$ be an integer sequence such that $$b(2^m(2n+1))=\sum\limit …

Let $a(n,m,k)$ be an integer sequence with e.g.f. $$A(x)=\operatorname{exp}\left(x + m\int\int (A(x))^k \, dx \, dx\right)$$ I don't know much about integrals, so here's a concrete example: $a(n,1,3) …

Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$. Let $a(n)$ be an integer sequence with generating function $\frac{1}{G(0)}$ where $$ G(j)=1-\frac{f(j)x}{G(j+1)} $$ Here we have $$ …

Let $a(n)$ be A266328 i.e. an integer sequence with exponential generating function $$ A(x)=\exp\int B(x) \,dx $$ such that $$ B(x)=\exp(-x)\exp\int A(x) \,dx $$ where the constant of integration is …

Let $a(n)$ be A112487 i.e. an integer sequence with exponential generating function $$ A(x)=\exp\left(\int (A(x)+A(x)^2)\,dx\right), \\ A(0)=1 $$ However, the definition in the name of the sequence i …

Let $a(n)$ be A006014 i.e. $$ a(n)=na(n-1)+\sum\limits_{j=1}^{n-2}a(j)a(n-j-1), \\ a(1)=1 $$ Also generating function $A(x)$ satisfies $$ A(x) = x(1 + A(x) + A(x)^2 + xA'(x)) $$ Let $$ R(n,q)=\sum\li …

Let $a(n)$ be A234289 i.e. integer sequence with exponential generating function $$ A(x)=1+A(x)^2\int \frac{1}{A(x)}\,dx $$ The sequence begins with $$ 1, 1, 3, 17, 147, 1729, 25827, 468593, 10012083 …

Let $a(n)$ be A235129 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies $$ A'(x) = 1 + A(x)\exp(A(x)) $$ The sequence begins with $$ 1, 1, 3, 12, 64, 424, 3358, …

Let $a(n)$ be A249833 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies $$ A(x) = 1 + \int A(x) + (A(x))^2\log A(x)\,dx $$ The sequence begins with $$ 1, 1, 2, 7 …

Let $a(n)$ be A036765 i.e. number of ordered rooted trees with $n$ non-root nodes and all outdegrees $\leqslant 3$. Here $$ a(n) = \frac{1}{n+1}\sum\limits_{j=0}^{\left\lfloor\frac{n}{2}\right\rfloor …

Let $a(n)$ be A007165 i.e. number of $P$-graphs with $2n$ edges. Here ordinary generating function $A(x)$ satisfies $$ A(x) = \frac{(1 + xA(x))(1 + 2xA(x))}{1 + 2xA(x) - (xA(x))^2} $$ Let $$ R(n, q) …

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 $f(n)$ be an arbitrary function. Let $a(n)$ be an integer sequence such that its ordinary generating function satisfies $$ A(x)=\sum\limits_{i=0}^{\infty}\frac{x^i}{\prod\limits_{j=0}^{i}(1-f(j) …

Please note that this question differs from one of the previous questions of mine. Let $f(n)$ be an arbitrary function with integer values. Let $c_n$ be an arbitrary integer sequence. Let $a(n)$ be a …