A128129 - OEIS

%I #15 Mar 12 2021 22:24:44

%S 1,2,4,7,12,20,32,50,76,114,168,244,350,496,696,967,1332,1820,2468,

%T 3324,4448,5916,7824,10292,13471,17548,22756,29384,37788,48408,61784,

%U 78578,99600,125838,158496,199036,249230,311224,387608,481506,596676

%N Expansion of (chi(-q^3)/ chi^3(-q) -1)/3 in powers of q where chi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

%H Vaclav Kotesovec, <a href="/A128129/b128129.txt">Table of n, a(n) for n = 1..10001</a>

%H Kevin Acres, David Broadhurst, <a href="https://arxiv.org/abs/1810.07478">Eta quotients and Rademacher sums</a>, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.

%H Andrew Sills, <a href="http://home.dimacs.rutgers.edu/~asills/EMDC/SillsEMDC-Rev.pdf">Towards an Automation of the Circle Method</a>, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S76.

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

%F Euler transform of period 18 sequence [ 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 0, ...].

%F G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)= u^2 -v -2*v^2 -4*u*v -6*u*v^2.

%F G.f. A(x) satisfies 0=f(A(x), A(x^3)) where f(u, v)= u^3 -v* (1+3*v+3*v^2)* (1+6*u+12*u^2).

%F a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (2^(7/4) * 3^(3/2) * n^(3/4)). - _Vaclav Kotesovec_, Jan 12 2017

%t nmax = 50; CoefficientList[Series[Product[(1 - x^(18*k))*(1 - x^(18*k - 3))*(1 - x^(18*k - 15))/((1 - x^(2*k - 1))*(1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 11 2017 *)

%o (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x^2+A)* eta(x^3+A)* eta(x^18+A)^2/ (eta(x^6+A)* eta(x^9+A)* eta(x+A)^2), n))}

%Y A128128(n)=3*a(n) if n>0.

%K nonn

%O 1,2

%A _Michael Somos_, Feb 15 2007