A365107 - OEIS

%I #4 Aug 24 2023 10:34:07

%S 1,0,1,1,18,101,1550,22492,424536,10283064,272319552,8959493401,

%T 328044534576,13799304374077,657306569855728,34694458662034731,

%U 2048559070407831424,132868259271772801185,9463476338179250300352,736376651361995115417850,62178423492630241909006224,5689134205956573233701281462

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

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

%t nmax = 21; CoefficientList[Series[Exp[Sum[x^Prime[k]/Prime[k]!^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2

%t 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}]

%Y Cf. A023998, A190476.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Aug 21 2023