A370541 Expansion of g.f. A(x) = Product_{n>=1} (1 + x^(n-1) + x^(2*n-1)) * (1 + x^n + x^(2*n-1)) * (1 - x^n - x^(2*n)).

2, 5, 0, -1, 0, -18, -15, -23, -36, -25, -52, -35, -42, -8, 13, 4, 96, 100, 208, 227, 388, 434, 499, 709, 670, 837, 883, 1057, 775, 1044, 819, 643, 535, -78, -345, -970, -1494, -3017, -3142, -5078, -6102, -7711, -9410, -11406, -13148, -15353, -17831, -18841, -22708, -22955, -26117

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

Cf. A000726, A319668.

Sequence in context: A100085 A159986 A065452 * A004598 A091086 A336686

Adjacent sequences: A370538 A370539 A370540 * A370542 A370543 A370544

Keyword

sign

Author

Paul D. Hanna, Feb 22 2024

Status

approved