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%I #19 Jan 31 2022 10:53:14
%S 1,1,31,3991,1340251,929043391,1153715889691,2333670966674671,
%T 7180487882511523051,31919495229412870788031,
%U 196909477461357591810377851,1632140626754602443266222263951,17701927686793740884448652685728651,245721757633690118910277310669218472671
%N E.g.f.: Sum_{n>=0} (1 - exp(-n^2*x))^n.
%H Seiichi Manyama, <a href="/A187755/b187755.txt">Table of n, a(n) for n = 0..162</a>
%F O.g.f.: Sum_{n>=0} n^(2*n) * n! * x^n / Product_{k=1..n} (1 + n^2*k*x).
%F a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(2*n) * k! * Stirling2(n,k).
%F a(n) == 1 (mod 10) for n>=0.
%F a(n) == 31 (mod 60) for n>=2.
%F a(n) ~ c * d^n * (n!)^3 / n, where d = 6.8312860494079582446988970296645779575650627187418208311407895492635... and c = 0.192038502554748256318271067254582378566365276592... . - _Vaclav Kotesovec_, May 08 2014
%e O.g.f.: F(x) = 1 + x + 31*x^2 + 3991*x^3 + 1340251*x^4 + 929043391*x^5 +...
%e where
%e F(x) = 1 + x/(1+x) + 2^4*2!*x^2/((1+2^2*1*x)*(1+2^2*2*x)) + 3^6*3!*x^3/((1+3^2*1*x)*(1+3^2*2*x)*(1+3^2*3*x)) + 4^8*4!*x^4/((1+4^2*1*x)*(1+4^2*2*x)*(1+4^2*3*x)*(1+4^2*4*x)) +...
%e ...
%e E.g.f.: A(x) = 1 + x + 31*x^2/2! + 3991*x^3/3! + 1340251*x^4/4! +...
%e where
%e A(x) = 1 + (1-exp(-x)) + (1-exp(-2^2*x))^2 + (1-exp(-3^2*x))^3 + (1-exp(-4^2*x))^4 + (1-exp(-5^2*x))^5 + (1-exp(-6^2*x))^6 +...
%t Flatten[{1,Table[Sum[(-1)^(n-k) * k^(2*n) * k! * StirlingS2[n,k], {k,0,n}], {n,1,20}]}] (* _Vaclav Kotesovec_, May 08 2014 *)
%o (PARI) {a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-k^2*x+x*O(x^n)))^k), n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=polcoeff(sum(m=0, n, m^(2*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) = sum(k=0, n, (-1)^(n-k)*k^(2*n)*k!*stirling(n, k, 2))}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A220181, A203798, A229260.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 03 2013
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