A360787 Expansion of Sum_{k>=0} x^k / (1 - (k*x)^2)^(k+1).
1, 1, 1, 3, 13, 40, 177, 965, 4733, 28103, 184065, 1191888, 8713549, 67005689, 528870257, 4526024267, 40051790333, 368513578472, 3583302492545, 35868588067501, 373781214260749, 4052932682659599, 45218033687522481, 523234757502985824, 6245693941097387773
Offset: 0
Programs
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Mathematica
Join[{1}, Table[Sum[Binomial[n-k,k] * (n-2*k)^(2*k), {k,0,n/2}], {n,1,30}]] (* Vaclav Kotesovec, Feb 21 2023 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k*x)^2)^(k+1))) -
PARI
a(n) = sum(k=0, n\2, (n-2*k)^(2*k)*binomial(n-k, k));
Formula
a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^(2*k) * binomial(n-k,k).