A132977 - OEIS

%I #16 Feb 16 2025 08:33:06

%S 1,2,5,12,26,50,92,168,295,496,818,1332,2126,3324,5126,7824,11793,

%T 17548,25857,37788,54734,78578,111968,158496,222842,311224,432095,

%U 596676,819504,1119624,1522282,2060448,2776514,3725294,4978142,6626988,8789042

%N Expansion of q^(-1/3) * (eta(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)))^2 in powers of q.

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

%H G. C. Greubel, <a href="/A132977/b132977.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="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q^(-2/3) * (chi(q) * chi(q^3))^2 * c(q^2) / (3 * b(q^2)) in powers of q where chi() is a Ramanujan theta function and b(), c() are cubic AGM functions.

%F Euler transform of period 12 sequence [ 2, 2, 4, 4, 2, -4, 2, 4, 4, 2, 2, 0, ...].

%F Expansion of (chi^3(q^3) / chi(q))^2 * (psi(-q^3) / psi(-q))^4 in powers of q where chi(), psi() are Ramanujan theta functions.

%F Expansion of q^(-1/3) * (eta(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)))^2 in powers of q.

%F G.f. = A112173(x) * A128758(x^2).

%F G.f.: (Product_{k>0} (1-x^(6*k))^4 / ( (1-x^k) * (1-x^(3*k)) * (1-x^(4*k)) * (1-x^(12*k)) ))^2.

%F a(n) = A132975(3*n + 1).

%F a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015

%e G.f. = 1 + 2*x + 5*x^2 + 12*x^3 + 26*x^4 + 50*x^5 + 92*x^6 + 168*x^7 + ...

%e G.f. = q + 2*q^4 + 5*q^7 + 12*q^10 + 26*q^13 + 50*q^16 + 92*q^19 + ...

%t nmax = 40; CoefficientList[Series[Product[((1-x^(6*k))^4 / ( (1-x^k) * (1-x^(3*k)) * (1-x^(4*k)) * (1-x^(12*k)) ))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)

%t a[ n_] := SeriesCoefficient[(QPochhammer[ x^6]^4 / (QPochhammer[ x] QPochhammer[ x^3] QPochhammer[ x^4] QPochhammer[ x^12]))^2, {x, 0, n}]; (* _Michael Somos_, Oct 31 2015 *)

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

%Y Cf. A112173, A128758, A132975.

%K nonn,changed

%O 0,2

%A _Michael Somos_, Sep 07 2007

%E Edited by R. J. Mathar and _N. J. A. Sloane_, Sep 01 2009