A246862 Expansion of phi(x) * f(x^3, x^5) in powers of x where phi(), f() are Ramanujan theta functions.

1, 2, 0, 1, 4, 1, 2, 2, 0, 4, 0, 0, 2, 0, 3, 2, 2, 0, 3, 4, 0, 2, 2, 2, 0, 2, 0, 2, 2, 0, 4, 0, 0, 1, 4, 0, 2, 2, 0, 5, 2, 2, 2, 4, 0, 0, 0, 0, 2, 4, 2, 0, 2, 0, 4, 2, 0, 0, 2, 0, 1, 2, 0, 2, 6, 0, 0, 4, 1, 8, 0, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 4, 0, 4, 2, 2

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 16 sequence [ 2, -3, 3, -1, 3, -4, 2, -2, 2, -4, 3, -1, 3, -3, 2, -2, ...].

Convolution of A000122 and A214264.

a(9*n + 2) = a(9*n + 8) = 0. a(9*n + 5) = A246863(n).

a(n) = A113407(2*n) = A226192(2*n) = A008441(4*n) = A134343(4*n) = A116604(8*n) = A125079(8*n) = A129447(8*n) = A138741(8*n).

Example

G.f. = 1 + 2*x + x^3 + 4*x^4 + x^5 + 2*x^6 + 2*x^7 + 4*x^9 + 2*x^12 + ...
G.f. = q + 2*q^17 + q^49 + 4*q^65 + q^81 + 2*q^97 + 2*q^113 + 4*q^145 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8] QPochhammer[ x^8], {x, 0, n}];

Prog

(PARI) {a(n) = if( n<0, 0, issquare(16 * n + 1) + 2 * sum(i=1, sqrtint(n), issquare(16 * (n - i^2) + 1)))};

Crossrefs

Cf. A000122, A008441, A113407, A116604, A125079, A129447, A134343, A138741, A214264, A226192, A246863.

Sequence in context: A258123 A121583 A228924 * A338773 A194686 A302996

Adjacent sequences: A246859 A246860 A246861 * A246863 A246864 A246865

Keyword

nonn

Author

Michael Somos, Sep 05 2014

Status

approved