A359920 a(n) = coefficient of x^n in A(x) such that x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

1, 1, 6, 29, 137, 690, 3815, 22579, 138353, 862692, 5451339, 34911444, 226475135, 1485571965, 9833401534, 65578882177, 440170565711, 2971402946711, 20161828468803, 137434420403678, 940701180157773, 6462787501335564, 44550102080595910, 308041365014677804, 2135938633975050831

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Eric Weisstein's World of Mathematics, Quintuple Product Identity.

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.

(1) x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

(2) 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) * (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 = 7.388458151593... and c = 0.36167254645... - Vaclav Kotesovec, Mar 19 2023

Formula (3) can be rewritten as the functional equation x = QPochhammer(x) * QPochhammer(y, x)/(1 - y) * QPochhammer(1/(x*y), x)/(1 - 1/(x*y)) * QPochhammer(y^2/x, x^2)/(1 - y^2/x) * QPochhammer(1/(x*y^2), x^2)/(1 - 1/(x*y^2)). - Vaclav Kotesovec, Jan 19 2024

Example

G.f.: A(x) = 1 + x + 6*x^2 + 29*x^3 + 137*x^4 + 690*x^5 + 3815*x^6 + 22579*x^7 + 138353*x^8 + 862692*x^9 + 5451339*x^10 + 34911444*x^11 + 226475135*x^12 + ...
where A = A(x) satisfies the doubly infinite sum
x = ... + x^12*(1/A^9 - A^8) + x^5*(1/A^6 - A^5) + x*(1/A^3 - A^2) + (1 - 1/A) + x^2*(A^3 - 1/A^4) + x^7*(A^6 - 1/A^7) + x^15*(A^9 - 1/A^10) + ... + x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)) + ...
also, by the Watson quintuple product identity,
x = (1-x)*(1-x*A)*(1-1/A)*(1-x*A^2)*(1-x/A^2) * (1-x^2)*(1-x^2*A)*(1-x/A)*(1-x^3*A^2)*(1-x^3/A^2) * (1-x^3)*(1-x^3*A)*(1-x^2/A)*(1-x^5*A^2)*(1-x^5/A^2) * (1-x^4)*(1-x^4*A)*(1-x^3/A)*(1-x^7*A^2)*(1-x^7/A^2) * ...
SPECIFIC VALUES.
A(x) at x = 100/738 diverges.
A(100/739) = 1.680090298639836342808608867776256534712736768391...
A(1/8) = 1.40048762211279862753069563580599076131617792526323...
A(1/9) = 1.28067125711115350114265686789651886973848631068277...

Mathematica

(* Calculation of constant d: *) With[{k = 1}, 1/r /. FindRoot[{r^3*s^3 * QPochhammer[r] * 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)) == k*r, 1/(-1 + s) + 1/(s*(-1 + r*s)) + (2*s)/(-r + s^2) - 2/(s - r*s^3) + (-QPolyGamma[0, -Log[r*s]/Log[r], r] + QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, -Log[r*s^2]/Log[r^2], r^2] + QPolyGamma[0, Log[s^2/r]/Log[r^2], r^2]) / (s*Log[r]) == 0}, {r, 1/7}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 18 2024 *)

Prog

(PARI) /* Using the doubly infinite series */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(x - sum(m=-#A, #A, (Ser(A)^(3*m) - 1/Ser(A)^(3*m+1)) * x^(m*(3*m+1)/2) ), #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) * (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. A359915, A359916, A359919, A359921, A359924, A359719.

Cf. A359914.

Sequence in context: A045445 A026884 A289801 * A110311 A030221 A271753

Adjacent sequences: A359917 A359918 A359919 * A359921 A359922 A359923

Keyword

nonn

Author

Paul D. Hanna, Jan 22 2023

Status

approved