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%I #16 Jan 31 2022 11:08:19
%S 1,1,31,117715,103060086811,35762522984596014091,
%T 7426384178532990386136937008571,
%U 1294894823429942167077615079432957932604708795,253092741940931724342360814670783323840910439695820558059377771,71352024076415778396125047299738711969981288880155877082517115026490261985701403851
%N E.g.f.: Sum_{n>=0} (1 - exp(-n^n*x))^n.
%H Seiichi Manyama, <a href="/A249584/b249584.txt">Table of n, a(n) for n = 0..26</a>
%F a(n) == 1 (mod 6) for n>=0.
%F O.g.f.: Sum_{n>=0} n^(n^2) * n! * x^n / Product_{k=1..n} (1 + n^n*k*x).
%F a(n) = Sum_{k=0..n} (-1)^(n-k) * (k^k)^n * k! * Stirling2(n,k).
%F a(n) ~ n! * n^(n^2). - _Vaclav Kotesovec_, Aug 24 2017
%e E.g.f.: A(x) = 1 + x + 31*x^2/2! + 117715*x^3/3! + 103060086811*x^4/4! +...
%e where
%e A(x) = 1 + (1-exp(-x)) + (1-exp(-2^2*x))^2 + (1-exp(-3^3*x))^3 + (1-exp(-4^4*x))^4 + (1-exp(-5^5*x))^5 + (1-exp(-6^6*x))^6 +...
%e Ordinary generating function.
%e O.g.f.: F(x) = 1 + x + 31*x^2 + 117715*x^3 + 103060086811*x^4 +...
%e where
%e F(x) = 1 + x/(1+x) + 2^4*2!*x^2/((1+2^2*1*x)*(1+2^2*2*x)) + 3^9*3!*x^3/((1+3^3*1*x)*(1+3^3*2*x)*(1+3^3*3*x)) + 4^16*4!*x^4/((1+4^4*1*x)*(1+4^4*2*x)*(1+4^4*3*x)*(1+4^4*4*x)) + 5^25*5!*x^5/((1+5^5*1*x)*(1+5^5*2*x)*(1+5^5*3*x)*(1+5^5*4*x)*(1+5^5*5*x)) +...
%t Flatten[{1, Table[Sum[(-1)^(n-k)*(k^k)^n*k!*StirlingS2[n, k], {k, 1, n}], {n, 1, 10}]}] (* _Vaclav Kotesovec_, Aug 24 2017 *)
%o (PARI) {a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-k^k*x +x*O(x^n)))^k), n)}
%o for(n=0, 15, print1(a(n), ", "))
%o (PARI) {a(n)=polcoeff(sum(m=0, n, m^(m^2)*m!*x^m/prod(k=1, m, 1+m^m*k*x +x*O(x^n))), n)}
%o for(n=0, 15, print1(a(n), ", "))
%o (PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*k^(k*n)*k!*stirling(n, k, 2))}
%o for(n=0, 10, print1(a(n), ", "))
%Y Cf. A220181, A187755, A203798.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 01 2014
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