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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 $a(n)$ 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 $C_n$ 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 $T(n,k)$ 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 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 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 $a(n)$ 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 ${n \brace k}$ 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 $G_n$ 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 …

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 $f(n)$ 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 $a(n)$ 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 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 A307389 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies $$ A(x)=\exp\left(\frac{\exp(2x)-2\exp(x)+2x+1}{2}\right) $$ The sequence begins with $$ …