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Expansion of Sum_{k>=0} (k * x/(1 - x))^k.
5

%I #16 Feb 18 2023 22:49:04

%S 1,1,5,36,350,4328,65132,1155904,23640724,547544032,14166236708,

%T 404944248104,12674392793900,431104742439088,15834117059443828,

%U 624575921756875960,26332801242942780668,1181750740315156943936,56244454481507648435012

%N Expansion of Sum_{k>=0} (k * x/(1 - x))^k.

%H Winston de Greef, <a href="/A355494/b355494.txt">Table of n, a(n) for n = 0..385</a>

%F a(n) = Sum_{k=1..n} k^k * binomial(n-1,k-1) for n > 0.

%F a(n) ~ exp(exp(-1)) * n^n. - _Vaclav Kotesovec_, Jul 05 2022

%t Join[{1}, Table[Sum[k^k * Binomial[n-1,k-1], {k,1,n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Jul 05 2022 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x/(1-x))^k))

%o (PARI) a(n) = if(n==0, 1, sum(k=1, n, k^k*binomial(n-1, k-1)));

%Y Cf. A355495, A355496.

%Y Cf. A086331.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jul 04 2022