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%I #15 Jul 09 2019 01:17:54
%S 1,1,3,1,25,1,721,1,6721,181441,151201,1,203575681,1,121080961,
%T 108972864001,3491282995201,1,133541574566401,1,304119455447808001,
%U 212878925715456001,309744468633601,1,17665560315112008499201,646300418472124416000001,841941782922240001
%N E.g.f.: Sum_{n>=0} x^n * (1 + x^n)^n / n!.
%C More generally, the following sums are equal:
%C (1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
%C (2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C here, q = x, p = 1, r = x. - _Paul D. Hanna_, Jul 03 2019
%C In general, let F(x) be a formal power series in x such that F(0)=1, then
%C Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! =
%C Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
%C here, F(x) = exp(x), q = , p = 1, r = x, m = 1. - _Paul D. Hanna_, Jul 06 2019
%H Paul D. Hanna, <a href="/A259208/b259208.txt">Table of n, a(n) for n = 0..1025</a>
%F E.g.f.: Sum_{n>=0} x^(n*(n+1)) * exp(x^(n+1)) / n!.
%F a(n) = Sum_{d|n} binomial(d, n/d - 1) * n!/d!.
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + x^3/3! + 25*x^4/4! + x^5/5! + 721*x^6/6! + x^7/7! + 6721*x^8/8! + 181441*x^9/9! +...
%e where
%e A(x) = 1 + x*(1+x) + x^2*(1+x^2)^2/2! + x^3*(1+x^3)^3/3! + x^4*(1+x^4)^4/4! + x^5*(1+x^5)^5/5! + x^6*(1+x^6)^6/6! +...
%e also
%e A(x) = exp(x) + x^2*exp(x^2) + x^6*exp(x^3)/2! + x^12*exp(x^4)/3! + x^20*exp(x^5)/4! + x^30*exp(x^6)/5! + x^42*exp(x^7)/6! +...
%e RELATED SERIES.
%e Below we illustrate the following identity at specific values of x:
%e Sum_{n>=0} x^n * (1 + x^n)^n / n! = Sum_{n>=0} x^(n*(n+1)) * exp(x^(n+1)) / n!.
%e (1) At x = 1/2, the following sums are equal
%e S1 = Sum_{n>=0} 2^(-n*(n+1)) * (2^n + 1)^n / n!,
%e S1 = Sum_{n>=0} 2^(-n*(n+1)) * exp( 1/2^(n+1) ) / n!,
%e where S1 = 1.97862370255774939923047215233920359461142155926360261512472...
%e (2) At x = 1/3, the following sums are equal
%e S2 = Sum_{n>=0} 3^(-n*(n+1)) * (3^n + 1)^n / n!,
%e S2 = Sum_{n>=0} 3^(-n*(n+1)) * exp( 1/3^(n+1) ) / n!,
%e where S2 = 1.52049327799122758174016893855657751957768595647287750170026...
%o (PARI) {a(n) = local(A=1); A = sum(m=0, n, x^m/m!*(1 + x^m +x*O(x^n))^m); n!*polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n) = local(A=1); A = sum(m=0, n, x^(m*(m+1)) / m! * exp(x^(m+1) +x*O(x^n))); n!*polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n) = if(n==0,1, sumdiv(n,d, binomial(d, n/d-1) * n!/d! ) )}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A326096.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 21 2015
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