A365107 Sum_{n>=0} a(n) * x^n / n!^2 = exp( Sum_{n>=1} x^prime(n) / prime(n)!^2 ).

1, 0, 1, 1, 18, 101, 1550, 22492, 424536, 10283064, 272319552, 8959493401, 328044534576, 13799304374077, 657306569855728, 34694458662034731, 2048559070407831424, 132868259271772801185, 9463476338179250300352, 736376651361995115417850, 62178423492630241909006224, 5689134205956573233701281462

Offset: 0

Author

Ilya Gutkovskiy, Aug 21 2023

Keywords

• nonn

Crossrefs

Cf. A023998, A190476.

Programs

Mathematica

nmax = 21; CoefficientList[Series[Exp[Sum[x^Prime[k]/Prime[k]!^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, Prime[k]]^2 Prime[k] a[n - Prime[k]], {k, 1, PrimePi[n]}]; Table[a[n], {n, 0, 21}]

Formula

a(0) = 1; a(n) = (1/n) * Sum_{p <= n, p prime} binomial(n,p)^2 * p * a(n-p).