A329258 Expansion of e.g.f. -log(1 - Sum_{k>=1} x^(k*(k + 1)/2) / (k*(k + 1)/2)!).

0, 1, 1, 3, 10, 44, 251, 1707, 13496, 122108, 1243201, 14060771, 174932274, 2374268974, 34910039164, 552782630401, 9378254813944, 169714311278784, 3263200704705648, 66434349885323328, 1427653109477475098, 32294539445483981821, 767051977023372086530

Offset: 0

Author

Ilya Gutkovskiy, Nov 09 2019

Keywords

• nonn

Crossrefs

Cf. A000629, A010054, A205803, A329259.

Programs

Mathematica

nmax = 22; CoefficientList[Series[-Log[1 - Sum[x^(k (k + 1)/2)/(k (k + 1)/2)!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Boole[IntegerQ[(8 n + 1)^(1/2)]] + Sum[Binomial[n, k] Boole[IntegerQ[(8 (n - k) + 1)^(1/2)]] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 22}]

Formula

a(0) = 0; a(n) = A010054(n) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * A010054(n-k) * k * a(k).