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A028572 Expansion of theta_3(z)*theta_3(2z) + theta_2(z)*theta_2(2z) in powers of q^(1/4). 1
1, 0, 0, 4, 2, 0, 0, 0, 2, 0, 0, 4, 4, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 8, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 4, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 4, 4, 0, 0, 0, 6, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 12, 2, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

LINKS

Table of n, a(n) for n=0..104.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of phi(x^4) * phi(x^8) + 4 * x^3 * psi(x^8) * psi(x^16) in powers of x where phi(), psi() are Ramanujan theta functions.

G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) (t/i) f(t) where q = exp(2 pi i t). - Michael Somos, Mar 23 2012

G.f.: Sum_{n,m} x^(3*(n^2 + m^2) + 2*n*m). - Michael Somos, Nov 20 2006

a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = 0. a(4*n) = A033715(n). a(8*n + 3) = 4 * A033761(n). - Michael Somos, Mar 23 2012

EXAMPLE

1 + 4*x^3 + 2*x^4 + 2*x^8 + 4*x^11 + 4*x^12 + 2*x^16 + 4*x^19 + 4*x^24 + ...

1 + 4*q^(3/4) +2*q +2*q^2 +4*q^(11/4) +4*q^3 +2*q^4 + 4*q^(19/4) +4*q^6 + ...

MATHEMATICA

terms = 105; max = Sqrt[terms] // Ceiling; s = Sum[x^(3*(n^2 + m^2) + 2*n*m), {n, -max, max}, {m, -max, max}]; CoefficientList[s, x][[1 ;; terms]] (* Jean-Fran├žois Alcover, Dec 03 2015, using 2nd g.f. *)

PROG

(PARI) {a(n) = if( n<1, n==0, qfrep( [3, 1; 1, 3], n)[n] * 2)} /* Michael Somos, Nov 20 2006 */

(PARI) {a(n) = if( n<1, n==0, if( n%4==1 || n%4==2, 0, 2 * sumdiv( n, d, kronecker( -2, d))))} /* Michael Somos, Mar 23 2012 */

CROSSREFS

Cf. A033715, A033761.

Sequence in context: A238012 A324802 A320647 * A107492 A159257 A258997

Adjacent sequences:  A028569 A028570 A028571 * A028573 A028574 A028575

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 8 12:31 EDT 2020. Contains 333314 sequences. (Running on oeis4.)