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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 Michael 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-François 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 STATUS approved

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Last modified April 21 13:29 EDT 2021. Contains 343153 sequences. (Running on oeis4.)