A260518 Expansion of psi(x)^2 * f(-x^3)^3 / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

1, 3, 5, 7, 8, 11, 13, 14, 17, 16, 21, 23, 25, 27, 21, 32, 33, 35, 37, 32, 42, 38, 45, 47, 40, 51, 56, 55, 50, 48, 61, 63, 64, 70, 56, 62, 73, 80, 77, 63, 81, 83, 74, 87, 72, 91, 98, 95, 104, 64, 101, 103, 105, 107, 88, 112, 98, 115, 114, 112, 121, 123, 125

Offset

0,2

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of psi(x) * psi(x^3) * f(x, x^2)^2 in powers of x where psi(), f() are Ramanujan theta functions.

Expansion of q^(-7/12) * eta(q^2)^4 * eta(q^3)^3 / eta(q)^3 in powers of q.

Euler transform of period 6 sequence [ 3, -1, 0, -1, 3, -4, ...].

G.f.: Product_{k>0} (1 - x^(2*k))^4 * (1 + x^k + x^(2*k))^3.

Example

G.f. = 1 + 3*x + 5*x^2 + 7*x^3 + 8*x^4 + 11*x^5 + 13*x^6 + 14*x^7 + ...
G.f. = q^7 + 3*q^19 + 5*q^31 + 7*q^43 + 8*q^55 + 11*q^67 + 13*q^79 + ...

Mathematica

a[ n_] := seriesCoefficient[ QPochhammer[ x^2]^4 QPochhammer[ x^3]^3 / QPochhammer[ x]^3, {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A)^3 / eta(x + A)^3, n))};
(PARI) q='q+O('q^99); Vec(eta(q^2)^4*eta(q^3)^3/eta(q)^3) \\ Altug Alkan, Aug 01 2018

Crossrefs

Sequence in context: A184415 A050111 A090542 * A190333 A190061 A288624

Adjacent sequences: A260515 A260516 A260517 * A260519 A260520 A260521

Keyword

nonn

Author

Michael Somos, Jul 28 2015

Status

approved