A261119 Expansion of f(x^2, -x^4) * f(x, x^5)^2 / f(-x^12, -x^12) in powers of x where f(, ) is Ramanujan's general theta function.

1, 2, 2, 2, 0, 0, 1, 2, 4, 0, 0, 0, 0, 4, 2, 2, 0, 0, 3, 2, 2, 2, 0, 0, 2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 3, 2, 4, 2, 0, 0, 0, 6, 2, 0, 0, 0, 0, 2, 4, 0, 0, 0, 2, 2, 2, 4, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 6, 2, 0, 0, 2, 4, 0, 2, 0, 0, 4, 4, 0, 0, 0, 0, 0, 4, 2

Offset: 0

Author

Michael Somos, Aug 08 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + x^6 + 2*x^7 + 4*x^8 + 4*x^13 + 2*x^14 + ...
G.f. = q^3 + 2*q^7 + 2*q^11 + 2*q^15 + q^27 + 2*q^31 + 4*q^35 + 4*q^55 + ...
		

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

Crossrefs

Cf. A000377, A129402, A192013, A257920, A257921, A259896, A261118.

Programs

Mathematica

a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 3}, (-1)^n DivisorSum[ m, KroneckerSymbol[ 12, #] KroneckerSymbol[ -2, m/#] &]]]; (* Michael Somos, Dec 22 2016 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^24 + A) / (eta(x + A)^2 * eta(x^6 + A)^2 * eta(x^8 + A)), n))};

PARI

a(n) = my(m = 4*n+3); (-1)^n*sumdiv(m, d, kronecker(12, d) * kronecker(-2, m/d)); \\ Michel Marcus, Dec 13 2017

Formula

Expansion of f(x^2, x^6) * f(x, x^5)^2 / f(x^4, x^8) in powers of x where f(,) is Ramanujan'sgeneral theta function.

Expansion of q^(-3/4) * eta(q^2)^3 * eta(q^3)^2 * eta(q^4) * eta(q^24) / (eta(q)^2 * eta(q^6)^2 * eta(q^8)) in powers of q.

Euler transform of period 24 sequence [ 2, -1, 0, -2, 2, -1, 2, -1, 0, -1, 2, -2, 2, -1, 0, -1, 2, -1, 2, -2, 0, -1, 2, -2, ...].

a(n) = (-1)^n * A257921(n) = A129402(2*n + 1) = A261118(3*n + 2) = A192013(4*n + 3) = A000377(4*n + 3).

a(2*n) = A257920(n). a(2*n + 1) = 2 * A259896(n). a(3*n) = A261118(n).