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A359916
a(n) = coefficient of x^n in A(x) such that A(x) = 1 + Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (1/A(x)^(3*n) - A(x)^(3*n+1)).
5
1, 1, 7, 48, 349, 2718, 22403, 192375, 1701544, 15389227, 141643233, 1322344998, 12491424723, 119177917679, 1146750961711, 11115577075944, 108437559699613, 1063849149587086, 10489551647580027, 103891138998923739, 1033113794091793406, 10310925888014393461
OFFSET
1,3
LINKS
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) A(x) = 1 + Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (1/A(x)^(3*n) - A(x)^(3*n+1)).
(2) -1 = Product_{n>=1} (1 - x^n) * (1 - x^n*A(x)) * (1 - x^n/A(x)) * (1 - x^(2*n-1)*A(x)^2) * (1 - x^(2*n-1)/A(x)^2), by the Watson quintuple product identity.
EXAMPLE
G.f.: A(x) = x + x^2 + 7*x^3 + 48*x^4 + 349*x^5 + 2718*x^6 + 22403*x^7 + 192375*x^8 + 1701544*x^9 + 15389227*x^10 + ...
where A = A(x) satisfies the doubly infinite series
A(x) - 1 = ... + x^12*(A^9 - 1/A^8) + x^5*(A^6 - 1/A^5) + x*(A^3 - 1/A^2) + (1 - A) + x^2*(1/A^3 - A^4) + x^7*(1/A^6 - A^7) + x^15*(1/A^9 - 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,
-1 = (1-x^1)*(1-x^1*A)*(1-x^1/A)*(1-x^1*A^2)*(1-x^1/A^2) * (1-x^2)*(1-x^2*A)*(1-x^2/A)*(1-x^3*A^2)*(1-x^3/A^2) * (1-x^3)*(1-x^3*A)*(1-x^3/A)*(1-x^5*A^2)*(1-x^5/A^2) * (1-x^4)*(1-x^4*A)*(1-x^4/A)*(1-x^7*A^2)*(1-x^7/A^2) * ...
PROG
(PARI) /* Using the doubly infinite series */
{a(n) = my(A=[0, 1, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(1 - Ser(A) + sum(m=-#A, #A, x^(m*(3*m+1)/2) * (1/Ser(A)^(3*m) - Ser(A)^(3*m+1)) ), #A-3) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* Using the quintuple product */
{a(n) = my(A=[0, 1, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(1 + prod(m=1, #A, (1 - x^m) * (1 - x^m*Ser(A)) * (1 - x^m/Ser(A)) * (1 - x^(2*m-1)*Ser(A)^2) * (1 - x^(2*m-1)/Ser(A)^2)), #A-3) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 22 2023
STATUS
approved