A260183 Expansion of f(x, x^2) * f(x^4, x^8) / f(-x^3, -x^6)^2 in powers of x where f(, ) is Ramanujan's general theta function.

1, 1, 1, 2, 3, 4, 6, 8, 10, 14, 18, 23, 30, 38, 47, 60, 74, 91, 114, 139, 169, 207, 250, 301, 364, 436, 520, 622, 739, 875, 1038, 1224, 1439, 1694, 1985, 2321, 2714, 3162, 3677, 4275, 4956, 5735, 6634, 7655, 8819, 10155, 11669, 13389, 15354, 17575, 20091

Offset

0,4

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(1/2) * eta(q^2) * eta(q^8) * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^6) * eta(q^24)) in powers of q.

Euler transform of period 24 sequence [ 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, ...].

a(n) = A003105(2*n + 1).

Example

G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 10*x^8 + 14*x^9 + ...
G.f. = 1/q + q + q^3 + 2*q^5 + 3*q^7 + 4*q^9 + 6*q^11 + 8*q^13 + 10*q^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^12] QPochhammer[ x^10, x^12] QPochhammer[ x^12, x^24] QPochhammer[ x^8] / QPochhammer[x], {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A)), n))};

Crossrefs

Cf. A003105.

Sequence in context: A053253 A095913 A102848 * A134157 A274194 A237751

Adjacent sequences: A260180 A260181 A260182 * A260184 A260185 A260186

Keyword

nonn

Author

Michael Somos, Nov 10 2015

Status

approved