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Let $a(n)$ be A225114 (i.e., number of skew partitions of $n$ whose diagrams have no empty rows and columns). Let $b(n)$ be an integer sequence with generating function $B(x)$ such that $$ B(x) = \cf …

Let $a(n)$ be A261041 (i.e., number of partitions of subsets of $\{1,2,\dotsc,n\}$, where consecutive integers are required to be in different parts). Let $b(n)$ be an integer sequence with generatin …

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 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 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 $G_n$ 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 $a(n)$ 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, …

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 …

Let $a(n)$ be A300483. Here $$ a(n) = 2\int\limits_{t \geqslant 0}T_n\left(\frac{t+1}{2}\right)\exp(-t)\,dt. $$ where $T_n(x)$ is $n$-th Chebyshev polynomial of first kind. Let $b(n)$ be an integer s …

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 ${n \brace k}$ 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 $f(n)$ 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 $a(n)$ 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 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 …