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%I #7 Feb 17 2024 08:18:08
%S 1,-3,9,-35,158,-784,4120,-22477,125884,-719090,4172414,-24521300,
%T 145659770,-873125958,5274788402,-32083526351,196312270733,
%U -1207543838993,7462752787850,-46315370172411,288536968531982,-1803738499481171,11311175379539531,-71135663269605615,448549432404511902
%N Expansion of g.f. A(x) satisfying Sum_{n>=0} (x^n + A(x))^n = Product_{n>=1} (1 - x^(2*n)) * (1 + x^n + A(x))^2 / (1 + x^(2*n) + A(x))^2.
%C Compare to identity: 1 + 2*Sum_{n>=0} x^(n^2) = Product_{n>=1} (1 - x^(2*n)) * (1 + x^n)^2 / (1 + x^(2*n))^2.
%H Paul D. Hanna, <a href="/A370341/b370341.txt">Table of n, a(n) for n = 1..225</a>
%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F (1) B(x) = Product_{n>=1} (1 - x^(2*n)) * (1 + x^n + A(x))^2 / (1 + x^(2*n) + A(x))^2.
%F (2) B(x) = Sum_{n>=0} (x^n + A(x))^n.
%F (3) B(x) = Sum_{n>=0} x^(n^2) / (1 - x^n*A(x))^(n+1).
%e G.f.: A(x) = x - 3*x^2 + 9*x^3 - 35*x^4 + 158*x^5 - 784*x^6 + 4120*x^7 - 22477*x^8 + 125884*x^9 - 719090*x^10 + 4172414*x^11 - 24521300*x^12 + ...
%e By definition, A = A(x) allows for the following expressions to equal
%e B(x) = 1 + (x + A) + (x^2 + A)^2 + (x^3 + A)^3 + (x^4 + A)^4 + (x^5 + A)^5 + ...
%e B(x) = (1 - x^2)*(1 + x + A)^2/(1 + x^2 + A)^2 * (1 - x^4)*(1 + x^2 + A)^2/(1 + x^4 + A)^2 * (1 - x^6)*(1 + x^3 + A)^2/(1 + x^6 + A)^2 * (1 - x^8)*(1 + x^4 + A)^2/(1 + x^8 + A)^2 * ...
%e where B(x) begins
%e B(x) = 1 + 2*x - 2*x^2 + 6*x^3 - 21*x^4 + 98*x^5 - 483*x^6 + 2514*x^7 - 13594*x^8 + 75624*x^9 - 429833*x^10 + 2484632*x^11 - 14559106*x^12 + ...
%o (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A=concat(A,0);
%o A[#A] = -polcoeff( sum(m=0,#A, (x^m + Ser(A))^m ) - prod(m=1,#A, (1 - x^(2*m))*(1 + x^m + Ser(A))^2/(1 + x^(2*m) + Ser(A))^2 ), #A-1)); A[n+1]}
%o for(n=1,30, print1(a(n),", "))
%Y Cf. A370341, A370342, A370344.
%K sign
%O 1,2
%A _Paul D. Hanna_, Feb 16 2024