A131963 Expansion of f(x, x^2) * f(x^4, x^12) in powers of x where f(, ) is Ramanujan's general theta function.

1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 0, 1, 2, 1, 1, 1, 1, 1, 0, 2, 0, 0, 1, 0, 2, 1, 3, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 1, 1, 2, 0, 0, 1, 1, 2, 0, 0, 2, 0, 1, 0, 1, 1, 2, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 3, 0, 1, 0, 0, 1, 2, 2, 0, 1, 1, 2

Offset

0,6

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

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

Expansion of q^(-13/24) * eta(q^2) * eta(q^3)^2 * eta(q^8)^2 / (eta(q) * eta(q^4) * eta(q^6)) in powers of q.

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

a(25*n + 13) = a(n). a(25*n + 3) = a(25*n + 8) = a(25*n + 18) = a(25*n + 23) = 0.

2 * a(n) = A123484(24*n + 13).

Example

G.f. = 1 + x + x^2 + x^4 + 2*x^5 + x^6 + x^7 + x^9 + x^11 + 2*x^12 + x^13 + ...
G.f. = q^13 + q^37 + q^61 + q^109 + 2*q^133 + q^157 + q^181 + q^229 + q^277 + ...

Mathematica

a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 13}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &] / 2]]; (* Michael Somos, Nov 04 2015 *)
a[ n_] := SeriesCoefficient[(1/2) x^(-1/2) EllipticTheta[ 4, 0, x^3] QPochhammer[ -x, x] EllipticTheta[ 2, 0, x^2], {x, 0, n}]; (* Michael Somos, Nov 04 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, n = 24*n + 13; sumdiv(n, d, kronecker( -12, d) * (n/d %2)) / 2)};
(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^8 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))};

Crossrefs

Cf. A123484.

Sequence in context: A081400 A378663 A328194 * A130538 A276007 A351619

Adjacent sequences: A131960 A131961 A131962 * A131964 A131965 A131966

Keyword

nonn

Author

Michael Somos, Aug 02 2007

Status

approved