A260158 Expansion of psi(x)^4 * psi(-x^3) / f(x) in powers of x where psi, f() are Ramanujan theta functions.

1, 3, 4, 6, 7, 6, 10, 12, 13, 15, 14, 18, 18, 21, 22, 18, 25, 27, 28, 24, 26, 33, 34, 42, 37, 30, 36, 42, 43, 45, 38, 48, 49, 42, 54, 42, 56, 57, 58, 60, 43, 63, 64, 66, 67, 63, 70, 60, 73, 84, 62, 78, 79, 72, 72, 66, 90, 87, 88, 90, 74, 78, 98, 96, 97, 78

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..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-5/6) * eta(q^2)^5 * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q)^3 * eta(q^6)) in powers of q.

Euler transform of period 12 sequence [ 3, -2, 2, -3, 3, -2, 3, -3, 2, -2, 3, -4, ...].

4 * a(n) = A260109(3*n + 2) = A124815(6*n + 5).

a(2*n + 1) = 3 * A260295(n).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/24) * exp(5*Pi/6) * 3^(1/4) * Pi^(3/2) * sqrt(2) * (3^(1/2)-1) / Gamma(7/12) / Gamma(2/3) / Gamma(3/4)^3 = A388906. - Simon Plouffe, Sep 21 2025

Example

G.f. = 1 + 3*x + 4*x^2 + 6*x^3 + 7*x^4 + 6*x^5 + 10*x^6 + 12*x^7 + 13*x^8 + ...
G.f. = q^7 + 3*q^23 + 4*q^39 + 6*q^55 + 7*q^71 + 6*q^87 + 10*q^103 + 12*q^119 + ...

Mathematica

a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 5}, DivisorSum[ m, m/# KroneckerSymbol[ 12, #]&] / 4]];
a[ n_] := SeriesCoefficient[ 2^(-9/2) x^(-7/8) EllipticTheta[ 2, 0, x^(1/2)]^4 EllipticTheta[ 2, Pi/4, x^(3/2)] / QPochhammer[ -x], {x, 0, n}];

Prog

(PARI) {a(n) = my(m); if( n<0, 0, m = 6*n + 5; sumdiv( m, d, m/d * kronecker( 12, d)) / 4)};
(PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^6 + A)), n))};

Crossrefs

Cf. A124815, A260109, A260295.

Sequence in context: A283740 A167161 A129000 * A390502 A388958 A317093

Adjacent sequences: A260155 A260156 A260157 * A260159 A260160 A260161

Keyword

nonn

Author

Michael Somos, Nov 09 2015

Status

approved