A347041 Stirling transform of pi (A000720).

0, 0, 1, 5, 21, 88, 389, 1852, 9525, 52632, 310141, 1936489, 12749204, 88149847, 637769490, 4812457992, 37763509549, 307453610201, 2592851608305, 22626572045811, 204197274002794, 1905132039608335, 18370391387293756, 183001650861913887, 1882207129695280320

Offset: 0

Author

Alois P. Heinz, Aug 13 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..575

Crossrefs

Cf. A000720, A008277, A048993, A230980, A307771.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0,
      numtheory[pi](m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);

Formula

G.f.: Sum_{k>=0} pi(k)*x^k / Product_{j=1..k} (1-j*x).

E.g.f.: Sum_{k>=0} pi(k)*(exp(x)-1)^k/k!.

a(n) = Sum_{k=0..n} Stirling2(n,k)*pi(k).