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%I #17 Feb 20 2023 07:36:28
%S 1,0,1,1,2,5,14,42,136,479,1825,7433,32053,145608,695081,3479117,
%T 18209842,99373513,563920590,3320674902,20255823092,127799984935,
%U 832807892861,5597481205009,38753768384761,276057156622776,2021100095469577,15193591060371577
%N Expansion of Sum_{k>=0} (x^2 / (1 - k*x))^k.
%H Winston de Greef, <a href="/A360708/b360708.txt">Table of n, a(n) for n = 0..630</a>
%F a(n) = Sum_{k=1..floor(n/2)} k^(n-2*k) * binomial(n-k-1,k-1) for n > 0.
%t Join[{1},Table[Sum[Binomial[n-k-1,k-1] * k^(n-2*k), {k,0,n/2}], {n,1,40}]] (* _Vaclav Kotesovec_, Feb 20 2023 *)
%o (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x^2/(1-k*x))^k))
%o (PARI) a(n) = if(n==0, 1, sum(k=1, n\2, k^(n-2*k)*binomial(n-k-1, k-1)));
%Y Cf. A080108, A360709.
%Y Cf. A000248, A360699.
%K nonn
%O 0,5
%A _Seiichi Manyama_, Feb 17 2023