A361052 Expansion of g.f. A(x) satisfying 4/x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

1, 4, 84, 2120, 61404, 1934548, 64379980, 2226478604, 79225597516, 2881791020120, 106672402111192, 4005192227754984, 152168779157569376, 5839221480075313396, 225986788425426186532, 8810672964167893735292, 345722424894740010814784, 13642862904817471637398044

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

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

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following.

(1) 4/x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

(2) 4/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.

(3) a(n) = Sum_{k=0..n-1} A361050(n,k) * 4^k, for n >= 1.

a(n) ~ c * d^n / n^(3/2), where d = 43.15078920061551630152405195461463024566432382819246... and c = 0.0036458304883879627950854318861022051996596920296... - Vaclav Kotesovec, Mar 19 2023

Example

G.f.: A(x) = x + 4*x^2 + 84*x^3 + 2120*x^4 + 61404*x^5 + 1934548*x^6 + 64379980*x^7 + 2226478604*x^8 + 79225597516*x^9 + ...
where A = A(x) satisfies the doubly infinite sum
4/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,
4/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) * ...

Prog

(PARI) /* Using the doubly infinite series */
{a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(4/x - sum(m=-#A, #A, (Ser(A)^(3*m) - 1/Ser(A)^(3*m+1)) * x^(m*(3*m+1)/2) ), #A-4) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* Using the quintuple product */
{a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(4/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-4) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A361050, A359921, A359924, A361051.

Sequence in context: A006344 A203244 A245544 * A053352 A322330 A356504

Adjacent sequences: A361049 A361050 A361051 * A361053 A361054 A361055

Keyword

nonn

Author

Paul D. Hanna, Mar 18 2023

Status

approved