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%I #13 Feb 16 2023 09:49:38
%S 1,1,2,10,77,808,11257,196072,4136897,103755904,3034193921,
%T 101901347944,3885951145969,166605168800704,7961498177012993,
%U 420976047757358776,24475992585921169553,1556007778666449968128,107625967130820901112833
%N Expansion of Sum_{k>=0} (x/(1 - k^2 * x))^k.
%F a(n) = Sum_{k=1..n} k^(2*(n-k)) * binomial(n-1,k-1) for n > 0.
%t Flatten[{1, Table[Sum[Binomial[n-1,k-1] * k^(2*(n-k)), {k,1,n}], {n,1,20}]}] (* _Vaclav Kotesovec_, Feb 16 2023 *)
%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-k^2*x))^k))
%o (PARI) a(n) = if(n==0, 1, sum(k=1, n, k^(2*(n-k))*binomial(n-1, k-1)));
%Y Cf. A080108, A135746, A234568, A355463, A355472.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Jul 03 2022