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A347041 Stirling transform of pi (A000720).

%I A347041 #10 Aug 13 2021 12:49:42
%S A347041 0,0,1,5,21,88,389,1852,9525,52632,310141,1936489,12749204,88149847,
%T A347041 637769490,4812457992,37763509549,307453610201,2592851608305,
%U A347041 22626572045811,204197274002794,1905132039608335,18370391387293756,183001650861913887,1882207129695280320
%N A347041 Stirling transform of pi (A000720).
%H A347041 Alois P. Heinz, <a href="/A347041/b347041.txt">Table of n, a(n) for n = 0..575</a>
%F A347041 G.f.: Sum_{k>=0} pi(k)*x^k / Product_{j=1..k} (1-j*x).
%F A347041 E.g.f.: Sum_{k>=0} pi(k)*(exp(x)-1)^k/k!.
%F A347041 a(n) = Sum_{k=0..n} Stirling2(n,k)*pi(k).
%p A347041 b:= proc(n, m) option remember; `if`(n=0,
%p A347041       numtheory[pi](m), m*b(n-1, m)+b(n-1, m+1))
%p A347041     end:
%p A347041 a:= n-> b(n, 0):
%p A347041 seq(a(n), n=0..27);
%Y A347041 Cf. A000720, A008277, A048993, A230980, A307771.
%K A347041 nonn
%O A347041 0,4
%A A347041 _Alois P. Heinz_, Aug 13 2021