A258764 Expansion of chi(-x^2) * psi(-x^3)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

1, 0, -1, -2, 0, 2, 0, 0, 0, 0, -1, 0, 2, 2, -2, 0, 1, 0, 0, -2, 0, 0, -2, 0, 0, 0, 0, -2, 2, 2, 0, 0, 1, 0, 0, -2, 0, 2, 0, 0, 0, 0, -1, -2, 2, 2, 0, 0, 2, 0, -2, 0, 0, 0, -2, 0, 0, 0, -2, -2, 0, 0, 0, 0, 2, 0, -1, 0, 0, 4, 0, 0, 0, 0, 0, -2, 0, 2, -2, 0, 3

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..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-2/3) * eta(q^2) * eta(q^3)^2 * eta(q^12)^2 / (eta(q^4) * eta(q^6)^2) in powers of q.

Euler transform of period 12 sequence [ 0, -1, -2, 0, 0, -1, 0, 0, -2, -1, 0, -2, ...].

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

a(n) = A129134(3*n + 2). -2 * a(n) = A082564(3*n + 2).

a(4*n) = A257399(n). a(8*n + 3) = -2 * A255318(n). a(8*n + 5) = 2 * A255319(n). a(8*n + 6) = -2 * A257402(n-1). a(16*n) = A257398(n). a(16*n + 2) = - A257399(n). a(16*n + 12) = 2 * A255317(n).

a(8*n + 1) = a(8*n + 7) = a(16*n + 4) = a(16*n + 8) = 0.

Example

G.f. = 1 - x^2 - 2*x^3 + 2*x^5 - x^10 + 2*x^12 + 2*x^13 - 2*x^14 + x^16 + ...
G.f. = q^2 - q^8 - 2*q^11 + 2*q^17 - q^32 + 2*q^38 + 2*q^41 - 2*q^44 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^4] QPochhammer[ x^3]^2 / QPochhammer[ x^6, x^12]^2, {x, 0, n}];
a[ n_] := If[ n < 0, 0, (-1)^Quotient[ n, 2] DivisorSum[ 3 n + 2, KroneckerSymbol[-2, #] &]];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x^4 + A) * eta(x^6 + A)^2), n))};
(PARI) {a(n) = if( n<0, 0, (-1)^(n\2) * sumdiv(3*n + 2, d, kronecker( -2, d)))};

Crossrefs

Cf. A082564, A129134, A255317, A255318, A255319, A257399, A257402.

Sequence in context: A335379 A358061 A342121 * A109083 A256812 A292251

Adjacent sequences: A258761 A258762 A258763 * A258765 A258766 A258767

Keyword

sign

Author

Michael Somos, Jun 09 2015

Status

approved