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A220353 G.f.: Sum_{n>=0} (1 - (1-x)^n)^n.

%I A220353 #27 Jul 01 2018 13:56:47
%S A220353 1,1,4,23,176,1697,19805,271669,4285195,76430799,1521161530,
%T A220353 33422603485,803584699252,20986514811397,591616582807036,
%U A220353 17905570068475471,579092313210791549,19931241131544637637,727395001560116046739,28057672464546863483509,1140566596105346550309751,48735378037084078566334897,2183719157723179429519093520,102386962560815561519635957007
%N A220353 G.f.: Sum_{n>=0} (1 - (1-x)^n)^n.
%C A220353 Limit n->infinity A220353(n)/A187826(n) = 1. - _Vaclav Kotesovec_, Nov 08 2014
%H A220353 Vaclav Kotesovec, <a href="/A220353/b220353.txt">Table of n, a(n) for n = 0..300</a>
%F A220353 G.f.: Sum_{n>=1} (1-x)^n * (1 - (1-x)^n)^(n-1).
%F A220353 a(n) = c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.93418651575946259471737... . - _Vaclav Kotesovec_, May 06 2014
%F A220353 In closed form, c = 2^(log(2)/2-1) / (log(2) * sqrt(Pi*(1-log(2)))). - _Vaclav Kotesovec_, May 03 2015
%e A220353 G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 176*x^4 + 1697*x^5 + 19805*x^6 +...
%e A220353 where the g.f. satisfies the identities:
%e A220353 (1) A(x) = 1 + x + (2*x - x^2)^2 + (3*x - 3*x^2 + x^3)^3 + (4*x - 6*x^2 + 4*x^3 - x^4)^4 + (5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5)^5 +...
%e A220353 (2) A(x) = (1-x) + (1-x)^2*(2*x - x^2) + (1-x)^3*(3*x - 3*x^2 + x^3)^2 + (1-x)^4*(4*x - 6*x^2 + 4*x^3 - x^4)^3 + (1-x)^5*(5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5)^4 +...
%t A220353 terms = 24;
%t A220353 gf = 1 + Sum[(1 - (1 - x)^n)^n, {n, 1, terms}] + O[x]^terms;
%t A220353 CoefficientList[gf, x] (* _Jean-François Alcover_, Jul 01 2018 *)
%o A220353 (PARI) {a(n)=local(q=1/(1-x+x*O(x^n)),A=1);A=sum(k=0,n,q^(-k^2)*(q^k-1)^k);polcoeff(A,n)}
%o A220353 for(n=0,30,print1(a(n),", "))
%o A220353 (PARI) {a(n)=local(q=1/(1-x+x*O(x^n)),A=1);A=sum(k=1,n+1,q^(-k^2)*(q^k-1)^(k-1));polcoeff(A,n)}
%o A220353 for(n=0,30,print1(a(n),", "))
%Y A220353 Cf. A220352, A122400, A187827, A187826.
%K A220353 nonn
%O A220353 0,3
%A A220353 _Paul D. Hanna_, Dec 11 2012
%E A220353 a(22)-a(23) corrected by _Andrew Howroyd_, Feb 22 2018