A164270 - OEIS

%I #11 Mar 12 2021 22:24:46

%S 1,-7,32,-114,350,-967,2468,-5916,13471,-29384,61784,-125838,249230,

%T -481506,909788,-1684824,3063657,-5478698,9648360,-16752522,28708214,

%U -48599047,81338660,-134687856,220802690,-358574468,577143752,-921144678,1458485460

%N Expansion of f(x^3)^3 * phi(x^3) / (f(x) * phi(x)^3) in powers of x where f(), phi() 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="/A164270/b164270.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^(-1/3) * eta(q)^7 * eta(q^4)^7 * eta(q^6)^14 / (eta(q^2)^18 * eta(q^3)^5 * eta(q^12)^5) in powers of q.

%F Euler transform of period 12 sequence [ -7, 11, -2, 4, -7, 2, -7, 4, -2, 11, -7, 0, ...].

%F a(n) = A164269(3*n + 1).

%e G.f. = 1 - 7*x + 32*x^2 - 114*x^3 + 350*x^4 - 967*x^5 + 2468*x^6 + ...

%e G.f. = q - 7*q^4 + 32*q^7 - 114*q^10 + 350*q^13 - 967*q^16 + 2468*q^19 + ...

%t f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164270[n_] := SeriesCoefficient[(f[q^3, -q^6]^3*f[q^3, q^3])/( (f[q, -q^2])*f[q, q]^3), {q, 0, n}]; Table[A164270[n], {n,0,50}] (* _G. C. Greubel_, Sep 16 2017 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^7 * eta(x^4 + A)^7 * eta(x^6 + A)^14 / (eta(x^2 + A)^18 * eta(x^3 + A)^5 * eta(x^12 + A)^5), n))};

%Y Cf. A164269.

%K sign

%O 0,2

%A _Michael Somos_, Aug 11 2009