A260308 Expansion of psi(x) * phi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

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

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 q^(-1/8) * eta(q^2)^2 * eta(q^6)^5 / (eta(q) * eta(q^3)^2 * eta(q^12)^2) in powers of q.

Euler transform of period 12 sequence [ 1, -1, 3, -1, 1, -4, 1, -1, 3, -1, 1, -2, ...].

a(n) = A259668(2*n) = A128580(4*n) = A129402(4*n) = A134177(4*n) = A190615(4*n) = A115660(8*n + 1) = A128581(8*n + 1) = A192013(8*n + 1).

Example

G.f. = 1 + x + 3*x^3 + 2*x^4 + 3*x^6 + 2*x^9 + x^10 + 2*x^12 + 4*x^13 + ...
G.f. = q + q^9 + 3*q^25 + 2*q^33 + 3*q^49 + 2*q^73 + q^81 + 2*q^97 + ...

Mathematica

a[ n_] := If[ n < 0, 0, DivisorSum[ 8 n + 1, KroneckerSymbol[ -6, #] &]];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # <= 3, Mod[#, 2], Mod[#, 24] > 12, 1 - Mod[#2, 2], True, (#2  + 1) KroneckerSymbol[ 3, #]^#2] & @@@ FactorInteger @ (8 n + 1))];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}];

Prog

(PARI) {a(n) = if( n<0, 0, sumdiv( 8*n + 1, d, kronecker( -6, d)))};
(PARI) {a(n) = my(A, p, e); if( n<0, 0, factor(8*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, 1, p%24>12, !(e%2), (e+1) * kronecker(3, p)^e)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^5 / (eta(x + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};

Crossrefs

Cf. A115660, A128580, A128581, A129402, A134177, A190615, A192013, A259668.

Sequence in context: A075115 A273528 A085080 * A079714 A286889 A286368

Adjacent sequences: A260305 A260306 A260307 * A260309 A260310 A260311

Keyword

nonn

Author

Michael Somos, Jul 22 2015

Status

approved