A262933 Expansion of f(-q^2, -q^5)^3 / (f(-q^1, -q^6) * f(-q^3, -q^4)^2) in powers of q where f(, ) is Ramanujan's general theta function.

1, 1, -2, 0, 5, -4, -7, 12, 4, -22, 7, 29, -26, -28, 52, 14, -82, 21, 106, -85, -105, 175, 53, -268, 70, 326, -264, -301, 505, 142, -742, 189, 885, -698, -805, 1323, 374, -1906, 483, 2205, -1732, -1946, 3185, 884, -4486, 1120, 5119, -3972, -4473, 7229, 2004

Offset

0,3

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 7 sequence [ 1, -3, 2, 2, -3, 1, 0, ...].

G.f.: T(q)/(T(q)-1), where T(q) = 1/q + 3 + 4*q + ... (cf. A108481). - Seiichi Manyama, Oct 11 2018

Example

G.f. = 1 + q - 2*q^2 + 5*q^4 - 4*q^5 - 7*q^6 + 12*q^7 + 4*q^8 - 22*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 3, -2, -2, 3, -1, 0}[[Mod[k, 7, 1]]], {k, n}], {x, 0, n}];
(* alternative program *)
QP:= QPochhammer; a[n_]:= SeriesCoefficient[(QP[q^2, q^7]*QP[q^5, q^7])^3/ (QP[q, q^7]*QP[q^6, q^7]*QP[q^3, q^7]^2*QP[q^4, q^7]^2), {q, 0, n}];
Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 18 2018 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[0, -1, 3, -2, -2, 3, -1][k%7 + 1]), n))};

Crossrefs

Cf. A108481.

Sequence in context: A378979 A083714 A215481 * A197253 A249693 A251420

Adjacent sequences: A262930 A262931 A262932 * A262934 A262935 A262936

Keyword

sign

Author

Michael Somos, Oct 04 2015

Status

approved