A131964 Expansion of f(x^2, x^10) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

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

Offset

0,4

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 phi(-x^4) * psi(-x^6) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.

Expansion of q^(-19/24) * eta(q^2) * eta(q^4)^2 * eta(q^6) * eta(q^24) / (eta(q) * eta(q^8) * eta(q^12)) in powers of q.

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

a(25*n + 19) = a(n). a(25*n + 4) = a(25*n + 9) = a(25*n + 14) = a(25*n + 24) = 0.

2 * a(n) = A123484(24*n + 19).

Example

G.f. = 1 + x + x^2 + 2*x^3 + x^5 + x^6 + x^8 + 2*x^10 + x^11 + x^12 + x^13 + ...
G.f. = q^19 + q^43 + q^67 + 2*q^91 + q^139 + q^163 + q^211 + 2*q^259 + q^283 + ...

Mathematica

a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 19}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &] / 2]]; (* Michael Somos, Nov 03 2015 *)
a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/4) EllipticTheta[ 4, 0, x^4] QPochhammer[ -x, x] EllipticTheta[ 2, Pi/4, x^3], {x, 0, n}]; (* Michael Somos, Nov 03 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, n = 24*n + 19; 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^4 + A)^2 * eta(x^6 + A) * eta(x^24 + A) / (eta(x + A) * eta(x^8 + A) * eta(x^12 + A)), n))};

Crossrefs

Cf. A123484.

Sequence in context: A363037 A330667 A116377 * A356241 A091430 A362221

Adjacent sequences: A131961 A131962 A131963 * A131965 A131966 A131967

Keyword

nonn

Author

Michael Somos, Aug 02 2007

Status

approved