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%I #5 Jun 20 2013 21:04:52
%S 1,1,1,1,31,151,451,1051,33601,663601,5187001,25905001,254322751,
%T 10408719751,128046088171,920598820051,29249420054401,723848667813601,
%U 12441294278905201,138598703861148241,4406639731521827551,93453608310743628151,1932981245635597160851,27744052310106087405451
%N E.g.f.: exp( Sum_{n>=1} sigma(n,n) * x^(n^2) / n^n ).
%C Here sigma(n,n) = A023887(n), the sum of the n-th powers of the divisors of n.
%C Compare to: exp( Sum_{n>=1} sigma(n)*x^n/n ), the g.f. of the partitions.
%F a(n) == 1 (mod 30) (conjecture - valid up to n=4000; if true for n>=0, why?).
%e E.g.f.: A(x) = 1 + x + x^2/2! + x^3/3! + 31*x^4/4! + 151*x^5/5! + 451*x^6/6! +...
%e where
%e log(A(x)) = x + 5*x^4/2^2 + 28*x^9/3^3 + 273*x^16/4^4 + 3126*x^25/5^5 + 47450*x^36/6^6 + 823544*x^49/7^7 +...+ A023887(n)*x^(n^2)/n^n +...
%o (PARI) {a(n)=n!*polcoeff(exp(sum(m=1,n,sigma(m,m)*(x^m/m)^m)+x*O(x^n)),n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A226838, A023887.
%K nonn
%O 0,5
%A _Paul D. Hanna_, Jun 20 2013
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