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%I #14 Jan 30 2022 11:40:12
%S 1,1,9,259,15789,1693771,287145789,71487432619,24798142070109,
%T 11518873418467051,6945333793188487869,5301472723402989073579,
%U 5018547949600497090304029,5790959348524892656227425131,8026963462960378548022418765949,13197920271743736945902641688868139
%N O.g.f.: Sum_{n>=0} n! * n^n * x^n / Product_{k=1..n} (1 - n^2*k*x).
%H Seiichi Manyama, <a href="/A229259/b229259.txt">Table of n, a(n) for n = 0..201</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^n.
%e O.g.f.: A(x) = 1 + x + 9*x^2 + 259*x^3 + 15789*x^4 + 1693771*x^5 +...
%e where
%e A(x) = 1 + x/(1-x) + 2!*2^2*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3!*3^3*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4!*4^4*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 + 9*x^2/2! + 259*x^3/3! + 15789*x^4/4! + 1693771*x^5/5! +...
%e where
%e E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/2^2 + (exp(9*x)-1)^3/3^3 + (exp(16*x)-1)^4/4^4 + (exp(25*x)-1)^5/5^5 +...
%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!*m^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^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, A229258, A229260, A229261, A229233, A229234, A220181, A122399.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 17 2013
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