A329256 Expansion of e.g.f. exp(Sum_{k>=1} x^(k^2) / (k^2)!).

1, 1, 1, 1, 2, 6, 16, 36, 106, 443, 1796, 6161, 23816, 122266, 643644, 2934296, 14002237, 83835433, 532282819, 3005258539, 17039094646, 115611682810, 848428608644, 5682350940168, 37297365940462, 281594230420802, 2323660209441962, 17929392395804072

Offset: 0

Author

Ilya Gutkovskiy, Nov 09 2019

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..611

Crossrefs

Cf. A000110, A010052, A205799, A205800, A205801, A205802, A205804, A353180.

Programs

Mathematica

nmax = 27; CoefficientList[Series[Exp[Sum[x^(k^2)/(k^2)!, {k, 1, Floor[nmax^(1/2)] + 1}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Boole[IntegerQ[k^(1/2)]] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 27}]

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, sqrtint(N), x^k^2/(k^2)!)))) \\ Seiichi Manyama, Apr 29 2022

PARI

a(n) = if(n==0, 1, sum(k=1, sqrtint(n), binomial(n-1, k^2-1)*a(n-k^2))); \\ Seiichi Manyama, Apr 29 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A010052(k) * a(n-k).