%I #5 Aug 24 2013 11:01:00
%S 1,1,4,24,252,3660,73560,1921080,63411600,2574406800,125747475840,
%T 7258472907840,487590023511360,37629962101892160,3299990581104497280,
%U 325758967714868688000,35904380354917794720000,4387164775718671231084800,590610815931660911894707200,87118296156852814044256665600
%N E.g.f.: exp( Sum_{n>=1} (1+x)^(n^2) * x^n/n ).
%C Compare the definition to: exp( Sum_{n>=1} (1+y)^(n^2) * x^n/n ), which yields an integer series whenever y is an integer.
%C Note that exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) * x^k ) yields an integer series (A206830).
%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 24*x^3/3! + 252*x^4/4! + 3660*x^5/5! +...
%e where, by definition,
%e log(A(x)) = (1+x)*x + (1+x)^4*x^2/2 + (1+x)^9*x^3/3 + (1+x)^16*x^4/4 + (1+x)^25*x^5/5+ (1+x)^36*x^6/6+ (1+x)^49*x^7/7 +...
%o (PARI) {a(n)=n!*polcoeff(exp(sum(m=1, n, (1+x)^(m^2)*x^m/m)+x*O(x^n)), n)}
%o for(n=0,25,print1(a(n),", "))
%Y Cf. A206830, A167006.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 24 2013