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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 18 2023
STATUS
approved