%I #14 Jan 30 2022 11:40:28

%S 1,1,3,31,573,18031,854613,57433951,5242645173,625589806831,

%T 95051257799973,17976303383444671,4153215615930529173,

%U 1154304694449774708751,380809177225169291456133,147420687475847638142996191,66303807316628093952943203573

%N O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 - n^2*k*x).

%H Seiichi Manyama, <a href="/A229258/b229258.txt">Table of n, a(n) for n = 0..227</a>

%F a(n) = Sum_{k=0..n} (k^2)^(n-k) * k! * Stirling2(n, k).

%F E.g.f.: Sum_{n>=0} (exp(n^2*x) - 1)^n / n^(2*n).

%e O.g.f.: A(x) = 1 + x + 3*x^2 + 31*x^3 + 573*x^4 + 18031*x^5 + 854613*x^6 +...

%e where

%e A(x) = 1 + x/(1-x) + 2!*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3!*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4!*x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...

%e Exponential Generating Function.

%e E.g.f.: E(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 573*x^4/4! + 18031*x^5/5! +...

%e where

%e E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/4^2 + (exp(9*x)-1)^3/9^3 + (exp(16*x)-1)^4/16^4 + (exp(25*x)-1)^5/25^5 +...

%t Flatten[{1,Table[Sum[(k^2)^(n-k) * k! * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, May 08 2014 *)

%o (PARI) {a(n)=polcoeff(sum(m=0,n,m!*x^m/prod(k=1,m,1-m^2*k*x +x*O(x^n))),n)}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {a(n)=n!*polcoeff(sum(m=0,n,(exp(m^2*x+x*O(x^n))-1)^m/m^(2*m)),n)}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {a(n)=sum(k=0, n, (k^2)^(n-k) * k! * stirling(n, k, 2))}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A229257, A229259, A229260, A229261; A229233, A229234, A220181, A122399.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 17 2013