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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).
1
1, 1, 5, 21, 90, 423, 2209, 12261, 69842, 403722, 2367829, 14096616, 85043323, 518567546, 3189349181, 19758783404, 123200215388, 772606927013, 4870004002571, 30837536428981, 196065919894270, 1251190368574657, 8011186350002373, 51451177669973807, 331365405433649972
OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Quintuple Product Identity.
FORMULA
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.
(1) x = P(x) * Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).
(2) x = P(x) * Sum_{n=-oo..+oo} x^(n*(3*n-1)/2) * A(x)^(3*n) * (x^n - 1/A(x)).
(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.
a(n) ~ c * d^n / n^(3/2), where d = 6.8614556596748153... and c = 0.3325444675505... - Vaclav Kotesovec, Mar 20 2023
EXAMPLE
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 + ...
MATHEMATICA
(* 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 *)
PROG
(PARI) /* Using the doubly infinite series */
{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);
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]}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Using the quintuple product */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
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]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A359920.
Sequence in context: A035011 A113987 A188707 * A333538 A164037 A218961
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 20 2023
STATUS
approved

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Last modified September 23 06:29 EDT 2024. Contains 376143 sequences. (Running on oeis4.)