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%I #3 Apr 30 2019 21:58:08
%S 1,1,1,4,22,139,1030,8445,74766,705916,7052531,74161365,817741825,
%T 9428968826,113449665799,1421957940220,18538429347923,251062976257349,
%U 3527558671120766,51360468675879418,774005747590247002,12059539535205433827,194049771204345008848,3221298357944442093597,55111256702479720092044,970767762280740604708491,17589320907639742684402262,327532084150359052415186316
%N G.f. A(x) satisfies: Sum_{n>=0} x^n * A(x)^(n^2) = Sum_{n>=0} x^n / [Product_{k=1..n} 1 - (2*k-1)*x].
%e G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 22*x^4 + 139*x^5 + 1030*x^6 + 8445*x^7 + 74766*x^8 + 705916*x^9 + 7052531*x^10 + 74161365*x^11 + 817741825*x^12 + ...
%e such that
%e B(x) = 1 + x*A(x) + x^2*A(x)^4 + x^3*A(x)^9 + x^4*A(x)^16 + x^5*A(x)^25 + x^6*A(x)^36 + x^7*A(x)^49 + x^8*A(x)^64 + x^9*A(x)^81 + ...
%e B(x) = 1 + x/(1-x) + x^2/((1-x)*(1-3*x)) + x^3/((1-x)*(1-3*x)*(1-5*x)) + x^4/((1-x)*(1-3*x)*(1-5*x)*(1-7*x)) + x^5/((1-x)*(1-3*x)*(1-5*x)*(1-7*x)*(1-9*x)) + x^6/((1-x)*(1-3*x)*(1-5*x)*(1-7*x)*(1-9*x)*(1-11*x)) + ....
%e where
%e B(x) = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 116*x^5 + 648*x^6 + 4088*x^7 + 28640*x^8 + 219920*x^9 + 1832224*x^10 + 16430176*x^11 + 157554048*x^12 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
%o A[#A] = polcoeff( sum(m=0, #A, x^m*( 1/prod(k=1,m,1-(2*k-1)*x +x*O(x^#A)) - Ser(A)^(m^2)) ), #A) ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A325452.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Apr 30 2019
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