OFFSET
0,1
COMMENTS
Consider function R(p,q,r) = Product_{n>=1} (1 + p^(n-1)*(q^n + r)) * (1 + p^n*(q^(n-1) + r)) * (1 - p^n*(q^n + r)) which yields Ramanujan's theta function at r = 0: R(p,q,0) = f(p,q) = Sum_{n=-oo..+oo} p^(n*(n-1)/2) * q^(n*(n+1)/2). This sequence arises from R(p,q,r) when p = x, q = x, and r = 1: A(x) = R(x,x,1).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..6400
EXAMPLE
G.f.: A(x) = 2 + 5*x - x^3 - 18*x^5 - 15*x^6 - 23*x^7 - 36*x^8 - 25*x^9 - 52*x^10 - 35*x^11 - 42*x^12 - 8*x^13 + 13*x^14 + 4*x^15 + 96*x^16 + 100*x^17 + ...
where A(x) equals the infinite product
A(x) = (2 + x)*(1 + 2*x)*(1 - x - x^2) * (1 + x + x^3)*(1 + x^2 + x^3)*(1 - x^2 - x^4) * (1 + x^2 + x^5)*(1 + x^3 + x^5)*(1 - x^3 - x^6) * (1 + x^3 + x^7)*(1 + x^4 + x^7)*(1 - x^4 - x^8) * (1 + x^4 + x^9)*(1 + x^5 + x^9)*(1 - x^5 - x^10) * (1 + x^5 + x^11)*(1 + x^6 + x^11)*(1 - x^6 - x^12) * (1 + x^6 + x^13)*(1 + x^7 + x^13)*(1 - x^7 - x^14) * ...
PROG
(PARI) {a(n) = my(A);
A = prod(m=1, n+1, (1 + x^(m-1) + x^(2*m-1)) * (1 + x^m + x^(2*m-1)) * (1 - x^m - x^(2*m)) +x*O(x^n));
polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Feb 22 2024
STATUS
approved