A353180 - OEIS

%I #20 May 01 2022 02:09:43

%S 1,1,2,6,25,130,810,5880,48790,455491,4725020,53915730,671141130,

%T 9050528630,131437406100,2045160117000,33944105995801,598591246152934,

%U 11176863039391538,220287874849834596,4570225746232479690,99557506547622369750,2272028399094852806100

%N Expansion of e.g.f. 1/(1 - Sum_{k>=1} x^(k^2) / (k^2)!).

%F a(0) = 1; a(n) = Sum_{k=1..floor(sqrt(n))} binomial(n,k^2) * a(n-k^2).

%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k^2] * a[n - k^2], {k, 1, Floor@Sqrt[n]}]; Array[a, 23, 0] (* _Amiram Eldar_, Apr 30 2022 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, sqrtint(N), x^k^2/(k^2)!))))

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, sqrtint(i), binomial(i, j^2)*v[i-j^2+1])); v;

%Y Cf. A006456, A205802, A329256, A353184.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Apr 29 2022