A260114 Expansion of f(x)^4 * phi(-x^3) / phi(-x) in powers of x where phi(), f() are Ramanujan theta functions.

1, 6, 14, 18, 21, 30, 38, 42, 43, 48, 62, 66, 74, 78, 64, 84, 98, 102, 110, 96, 133, 126, 108, 138, 112, 150, 158, 162, 183, 126, 182, 192, 194, 198, 160, 210, 180, 222, 230, 192, 242, 252, 288, 228, 208, 270, 278, 282, 273, 240, 252, 306, 314, 336, 294, 330

Offset: 0

Author

Michael Somos, Jul 16 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 6*x + 14*x^2 + 18*x^3 + 21*x^4 + 30*x^5 + 38*x^6 + 42*x^7 + ...
G.f. = q + 6*q^7 + 14*q^13 + 18*q^19 + 21*q^25 + 30*q^31 + 38*q^37 + ...
		

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

Crossrefs

Cf. A113421, A124815, A260518.

Programs

Mathematica

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

PARI

{a(n) = my(m = 6*n + 1); if (n<0, 0, sumdiv( m, d, d * kronecker( -3, d) * kronecker( -4, m/d)))};

PARI

{a(n) = my(m = 6*n + 1); if (n<0, 0, sumdiv( m, d, m/d * kronecker( 12, d)))};

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^13 * eta(x^3 + A)^2 / (eta(x + A)^6 * eta(x^4 + A)^4 * eta(x^6 + A)), n))};

Formula

Expansion of q^(-1/6) * eta(q^2)^13 * eta(q^3)^2 / (eta(q)^6 * eta(q^4)^4 * eta(q^6)) in powers of q.

Euler transform of period 12 sequence [ 6, -7, 4, -3, 6, -8, 6, -3, 4, -7, 6, -4, ...].

a(n) = A113421(6*n + 1) = A124815(6*n + 1).

a(2*n + 1) = 6 * A260518(n). - Michael Somos, Oct 07 2015