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%I #22 Jan 31 2022 11:04:11
%S 1,15,1267,316275,174397531,179770837155,310789895286907,
%T 834906367019076675,3293344593080631993211,18259284528276047000517795,
%U 137429981152689382429349060347,1365009985652048448232840864764675,17475885712645599218827214639383437691
%N E.g.f.: Sum_{n>=1} (1 - exp(-n^2*x))^n / n.
%C Compare to the trivial identity: x = Sum_{n>=1} (1 - exp(-x))^n/n.
%C Compare to the e.g.f. of A092552: Sum_{n>=1} (1 - exp(-n*x))^n/n.
%H Seiichi Manyama, <a href="/A220179/b220179.txt">Table of n, a(n) for n = 1..162</a>
%F O.g.f.: Sum_{n>=1} n^(2*n-1) * n! * x^n / Product_{k=1..n} (1 - n^2*k*x). - _Paul D. Hanna_, Jan 05 2013
%F a(n) = Sum_{k=1..n} (-1)^(n-k) * k^(2*n-1) * k! * Stirling2(n,k). - _Paul D. Hanna_, Jan 05 2013
%F a(n) ~ c * d^n * (n!)^3 / n^2, where d = 6.8312860494079582446988970296645779575650627187418208311407895492635... and c = 0.175744118254830086361220160145768507562830495967... . - _Vaclav Kotesovec_, May 08 2014
%e E.g.f.: A(x) = x + 15*x^2/2! + 1267*x^3/3! + 316275*x^4/4! + 174397531*x^5/5! +...
%e where
%e A(x) = (1-exp(-x)) + (1-exp(-4*x))^2/2 + (1-exp(-9*x))^3/3 + (1-exp(-16*x))^4/4 + (1-exp(-25*x))^5/5 +...
%t Table[Sum[(-1)^(n-k) * k^(2*n-1) * k! * StirlingS2[n,k], {k,1,n}], {n,1,20}] (* _Vaclav Kotesovec_, May 08 2014 *)
%o (PARI) a(n)=n!*polcoeff(sum(k=1, n, (1-exp(-k^2*x+x*O(x^n)))^k/k), n)
%o for(n=1,20,print1(a(n),", "))
%o (PARI) a(n)=polcoeff(sum(m=1, n, m^(2*m-1)*m!*x^m/prod(k=1, m, 1+m^2*k*x+x*O(x^n))), n) \\ _Paul D. Hanna_, Jan 05 2013
%o for(n=1,20,print1(a(n),", "))
%o (PARI) {a(n)=sum(k=1, n, (-1)^(n-k)*k^(2*n-1)*k!*stirling(n, k, 2))}
%o for(n=1, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Jan 05 2013
%Y Cf. A092552, A187755, A220181, A242228.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Dec 06 2012
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