A329256 - OEIS

%I #14 May 01 2022 06:02:24

%S 1,1,1,1,2,6,16,36,106,443,1796,6161,23816,122266,643644,2934296,

%T 14002237,83835433,532282819,3005258539,17039094646,115611682810,

%U 848428608644,5682350940168,37297365940462,281594230420802,2323660209441962,17929392395804072

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

%H Seiichi Manyama, <a href="/A329256/b329256.txt">Table of n, a(n) for n = 0..611</a>

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

%t 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]!

%t 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}]

%o (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

%o (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

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

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Nov 09 2019