A213023 - OEIS

%I #12 Mar 12 2021 22:24:46

%S 1,2,2,3,2,2,4,4,5,3,4,5,4,6,4,4,5,7,5,3,6,8,8,8,6,3,7,6,10,6,5,10,4,

%T 8,7,8,10,6,9,8,5,10,10,11,6,9,11,6,12,9,8,8,10,9,6,6,15,12,9,9,6,13,

%U 10,13,10,7,14,12,12,8,7,13,10,16,9,10,10,12

%N Expansion of psi(x)^2 * psi(-x^3) / chi(-x^2) in powers of x where psi(), chi() are Ramanujan theta functions.

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

%H G. C. Greubel, <a href="/A213023/b213023.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q^(-17/24) * eta(q^2)^3 * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q)^2 * eta(q^6)) in powers of q.

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

%F a(n) = A180312(3*n + 1).

%e 1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 3*x^9 + ...

%e q^17 + 2*q^41 + 2*q^65 + 3*q^89 + 2*q^113 + 2*q^137 + 4*q^161 + 4*q^185 + ...

%t QP := QPochhammer; a[n_]:=SeriesCoefficient[(QP[q^2]^3*QP[q^3]*QP[q^4] *QP[q^12])/(QP[q]^2*QP[q^6]), {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Jan 07 2018 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A)^2 * eta(x^6 + A)), n))}

%Y Cf. A180312.

%K nonn

%O 0,2

%A _Michael Somos_, Jun 03 2012