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%I #13 Jan 31 2024 06:14:37
%S 1,1,3,15,80,466,2863,18265,119856,803815,5485100,37962429,265843161,
%T 1880196279,13411084410,96363049074,696851538230,5067830579847,
%U 37040831752291,271947326550486,2004646999165517,14831053502690715,110088540151473330,819638482252323929,6119312652926134529
%N Expansion of g.f. A(x) satisfying -x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^(n^2).
%H Paul D. Hanna, <a href="/A369085/b369085.txt">Table of n, a(n) for n = 0..400</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F (1) -x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^(n^2).
%F (2) 1-x = Sum_{n=0..+oo} (-1)^n * (1 + x^n) * x^(n*(n-1)/2) * A(x)^(n^2).
%F (3) -x = Product_{n>=1} (1 - x^n*A(x)^(2*n)) * (1 - x^(n-1)*A(x)^(2*n-1)) * (1 - x^n*A(x)^(2*n-1)), by the Jacobi triple product identity.
%F From _Vaclav Kotesovec_, Jan 31 2024: (Start)
%F Formula (3) can be rewritten as the functional equation QPochhammer(x*y^2) * QPochhammer(1/(x*y), x*y^2)/(1 - 1/(x*y)) * QPochhammer(1/y, x*y^2)/(1 - 1/y) = -x.
%F a(n) ~ c * d^n / n^(3/2), where d = 7.95151530799625227004980814233852982518684056958755... and c = 0.179612489322086717165885798939686030305429728101... (End)
%e G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 80*x^4 + 466*x^5 + 2863*x^6 + 18265*x^7 + 119856*x^8 + 803815*x^9 + 5485100*x^10 + 37962429*x^11 + 265843161*x^12 + ...
%e where
%e -x = ... - x^6*A(x)^9 + x^3*A(x)^4 - x*A(x) + 1 - A(x) + x*A(x)^4 - x^3*A(x)^9 + x^6*A(x)^16 - x^10*A(x)^25 + x^15*A(x)^36 +- ... + (-1)^n * x^(n*(n-1)/2) * A(x)^(n^2) + ...
%t (* Calculation of constant d: *) 1/r /. FindRoot[{1 + s^2*QPochhammer[1/s, r*s^2]*QPochhammer[1/(r*s), r*s^2] * QPochhammer[r*s^2, r*s^2] / ((-1 + s)*(-1 + r*s)) == 0, (-2 + s + r*s)/s^2 - 2*r*(-1 + s)*(-1 + r*s) * Derivative[0, 1][QPochhammer][1/s, r*s^2]/ QPochhammer[1/s, r*s^2] + 2*r*s^2*QPochhammer[1/s, r*s^2]*QPochhammer[r*s^2, r*s^2] * Derivative[0, 1][QPochhammer][1/(r*s), r*s^2] + (-1 + s)*(-1 + r*s)*(-(-2*QPolyGamma[0, 1, r*s^2] + QPolyGamma[0, Log[1/s]/Log[r*s^2], r*s^2] + QPolyGamma[0, Log[1/(r*s)]/Log[r*s^2], r*s^2])/(s^2* Log[r*s^2]) - 2*r*Derivative[0, 1][QPochhammer][r*s^2, r*s^2] / QPochhammer[r*s^2, r*s^2]) == 0}, {r, 1/8}, {s, 3/2}, WorkingPrecision -> 70] (* _Vaclav Kotesovec_, Jan 31 2024 *)
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); M=ceil(sqrt(2*n));
%o A[#A] = polcoeff(x + sum(m=-M, M, (-1)^m * x^(m*(m-1)/2) * Ser(A)^(m^2)), #A-1)); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A355349.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 22 2024
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