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Define a transformation $T_s$ of integer sequence ${ a_n }$ by $$b_n=T_s(a_n)={n \choose s} \sum_{i=s}^{n-1} \frac{a_i}{{n frac{a_i}{{i \choose s}},$$ for a fixed $s \in \mathbb{N}.$

For instance, if we aplly the transformation $T_2$ to the sequence $a_n=1$ then we get the sequence $b_n=n(n-2).$ Maple code for the sample

T:=(a,s)->factor(simplify(product(n-i,i=0..s-1)*sum(a(k)/product(k-i,i=0..s-1),k=s..n-1)));
a:=n->1:T(a,2);
n(n-2)

Question. Suppose that $a_n$ has a generating function (ordinary or exponential or another one) $a(x).$ What is the generating function of transformed sequence?

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Define a transformation $T_s$ of integer sequence ${ a_n }$ by $$b_n=T_s(a_n)={n \choose s} \sum_{i=s}^{n-1} \frac{a_i}{{n \choose s}},$$ for a fixed $s \in \mathbb{N}.$

For instance, if we aplly the transformation $T_2$ to the sequence $a_n=1$ then we get the sequence $b_n=n(n-2).$ Maple code for the sample

a:=n->1

T:=(a,s)->factor(simplify(product(n-i,i=0..s-1)*sum(a(k)/product(k-i,i=0..s-1),k=s..n-1)));
a:=n->1:T(a,2);
n(n-2)

Question. Suppose that $a_n$ has a generating function (ordinary or exponential or another one) $a(x).$ What is the generating function of transformed sequence?

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# Find generating function

Define a transformation $T_s$ of integer sequence ${ a_n }$ by $$b_n=T_s(a_n)={n \choose s} \sum_{i=s}^{n-1} \frac{a_i}{{n \choose s}},$$ for a fixed $s \in \mathbb{N}.$

For instance, if we aplly the transformation $T_2$ to the sequence $a_n=1$ then we get the sequence $b_n=n(n-2).$ Maple code for the sample

a:=n->1
T:=(a,s)->factor(simplify(product(n-i,i=0..s-1)*sum(a(k)/product(k-i,i=0..s-1),k=s..n-1)));
a:=n->1:T(a,2);
n(n-2)

Question. Suppose that $a_n$ has a generating function (ordinary or exponential or another one) $a(x).$ What is the generating function of transformed sequence?