A245432 Expansion of f(-q^3, -q^5)^2 / (psi(-q) * phi(q^2)) in powers of q where phi(), psi(), f() are Ramanujan theta functions.

1, 1, -1, -2, 3, 4, -6, -8, 11, 15, -20, -26, 34, 44, -56, -72, 91, 114, -143, -178, 220, 272, -334, -408, 498, 605, -732, -884, 1064, 1276, -1528, -1824, 2171, 2580, -3058, -3616, 4269, 5028, -5910, -6936, 8124, 9498, -11088, -12922, 15034, 17468, -20264

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 (f(-q^3, -q^5) / f(-q^1, -q^7)) * (psi(q^4) / phi(q^2)) in powers of q where phi(), psi(), f() are Ramanujan theta functions.

Euler transform of period 8 sequence [ 1, -2, -1, 4, -1, -2, 1, 0, ...].

Convolution quotient of A244526 and A226192.

a(n) = (-1)^floor(n/2) * A115671(n).

a(n) = A224216(n) unless n=0. a(2*n+1) = A210063(n).

Example

G.f. = 1 + q - q^2 - 2*q^3 + 3*q^4 + 4*q^5 - 6*q^6 - 8*q^7 + 11*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2, q^4] / QPochhammer[ q^4, q^8]^2)^2 QPochhammer[ q^3, q^8] QPochhammer[ q^5, q^8] / (QPochhammer[ q^1, q^8] QPochhammer[ q^7, q^8]), {q, 0, n}];

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[0, -1, 2, 1, -4, 1, 2, -1][k%8 + 1]), n))};
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); (-1)^(n \ 2) * polcoeff( (1 + eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A))) / 2, n))};

Crossrefs

Cf. A115671, A210063, A226192, A224216, A244526.

Sequence in context: A133153 A100673 A224216 * A115671 A208856 A105782

Adjacent sequences: A245429 A245430 A245431 * A245433 A245434 A245435

Keyword

sign

Author

Michael Somos, Jul 21 2014

Status

approved