A329261 Expansion of e.g.f. -log(1 - Sum_{k>=1} x^prime(k) / prime(k)!).

0, 0, 1, 1, 3, 11, 40, 232, 1246, 8912, 65766, 561001, 5198424, 52513111, 577791292, 6806860347, 86303601008, 1163845620633, 16701819148776, 253608108810052, 4065574363467636, 68608467057149112, 1215544196988580438, 22564088376584800717

Offset: 0

Author

Ilya Gutkovskiy, Nov 09 2019

Keywords

• nonn

Crossrefs

Cf. A000629, A010051, A190476, A305618, A305619.

Programs

Mathematica

nmax = 23; CoefficientList[Series[-Log[1 - Sum[x^Prime[k]/Prime[k]!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Boole[PrimeQ[n]] + Sum[Binomial[n, k] Boole[PrimeQ[n - k]] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 23}]

Formula

a(0) = 0; a(n) = A010051(n) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * A010051(n-k) * k * a(k).