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A360580 Expansion of g.f. A(x) satisfying x = P(x) * Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)), where P(x) = 1/Product_{n>=1} (1 - x^n).

%I A360580 #13 Jan 19 2024 04:09:20
%S A360580 1,1,5,21,90,423,2209,12261,69842,403722,2367829,14096616,85043323,
%T A360580 518567546,3189349181,19758783404,123200215388,772606927013,
%U A360580 4870004002571,30837536428981,196065919894270,1251190368574657,8011186350002373,51451177669973807,331365405433649972
%N A360580 Expansion of g.f. A(x) satisfying x = P(x) * Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)), where P(x) = 1/Product_{n>=1} (1 - x^n).
%H A360580 Paul D. Hanna, <a href="/A360580/b360580.txt">Table of n, a(n) for n = 0..300</a>
%H A360580 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>.
%F A360580 G.f. A(x) = Sum_{n>=0} a(n)*x^n, along with P(x) = 1/Product_{n>=1} (1 - x^n), satisfies the following.
%F A360580 (1) x = P(x) * Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).
%F A360580 (2) x = P(x) * Sum_{n=-oo..+oo} x^(n*(3*n-1)/2) * A(x)^(3*n) * (x^n - 1/A(x)).
%F A360580 (3) x = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^(2*n-1)*A(x)^2) * (1 - x^(2*n-1)/A(x)^2), by the Watson quintuple product identity.
%F A360580 a(n) ~ c * d^n / n^(3/2), where d = 6.8614556596748153... and c = 0.3325444675505... - _Vaclav Kotesovec_, Mar 20 2023
%e A360580 G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 90*x^4 + 423*x^5 + 2209*x^6 + 12261*x^7 + 69842*x^8 + 403722*x^9 + 2367829*x^10 + ...
%t A360580 (* Calculation of constant d: *) 1/r /. FindRoot[{r^3*s^3 * QPochhammer[1/(r*s^2), r^2] * QPochhammer[1/(r*s), r] * QPochhammer[s, r] * QPochhammer[s^2/r, r^2] / ((-1 + s)*(-1 + r*s)*(-r + s^2)*(-1 + r*s^2)) == r, 1/(-1 + s) + 1/(s*(-1 + r*s)) + 2*s/(-r + s^2) - 2/(s - r*s^3) + (QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, -1/2 - Log[s]/Log[r], r^2] + QPolyGamma[0, -1/2 + Log[s]/Log[r], r^2] - QPolyGamma[0, -Log[r*s]/Log[r], r])/(s*Log[r]) == 0}, {r, 1/6}, {s, 2}, WorkingPrecision -> 70] (* _Vaclav Kotesovec_, Jan 19 2024 *)
%o A360580 (PARI) /* Using the doubly infinite series */
%o A360580 {a(n) = my(A=[1],P = 1/prod(m=1,n,1-x^m +x*O(x^n))); for(i=1, n, A = concat(A, 0);
%o A360580 A[#A] = polcoeff(x - P*sum(m=-#A, #A, x^(m*(3*m-1)/2) * Ser(A)^(3*m-1) * (x^m*Ser(A) - 1) ) , #A-1) ); A[n+1]}
%o A360580 for(n=0, 30, print1(a(n), ", "))
%o A360580 (PARI) /* Using the quintuple product */
%o A360580 {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
%o A360580 A[#A] = polcoeff(x - prod(m=1, #A, (1 - x^m*Ser(A)) * (1 - x^(m-1)/Ser(A)) * (1 - x^(2*m-1)*Ser(A)^2) * (1 - x^(2*m-1)/Ser(A)^2) ), #A-1) ); A[n+1]}
%o A360580 for(n=0, 30, print1(a(n), ", "))
%Y A360580 Cf. A359920.
%K A360580 nonn
%O A360580 0,3
%A A360580 _Paul D. Hanna_, Mar 20 2023