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A182031 Expansion of q^(-5/24) * (eta(q^3) * eta(q^6))^3 / (eta(q) * eta(q^2))^4 in powers of q.

%I A182031 #19 Feb 16 2025 08:33:13
%S A182031 1,4,18,53,163,414,1059,2431,5553,11844,25013,50391,100362,193136,
%T A182031 367371,680705,1247247,2238408,3975218,6941384,12003156,20465599,
%U A182031 34581525,57737205,95601892,156665029,254777220,410580026,657015874
%N A182031 Expansion of q^(-5/24) * (eta(q^3) * eta(q^6))^3 / (eta(q) * eta(q^2))^4 in powers of q.
%C A182031 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A182031 H.-C. Chan, On the Andrews-Schur proof of the Rogers-Ramanujan identities, Ramanujan J. 23 (2010), no. 1-3, 417-431. see p. 430 Theorem 7.
%H A182031 G. C. Greubel, <a href="/A182031/b182031.txt">Table of n, a(n) for n = 0..1000</a>
%H A182031 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A182031 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A182031 Expansion of (psi(x^3) * phi(-x^3))^3 / (psi(x) * phi(-x))^4 in powers of x where phi(), psi() are Ramanujan theta functions.
%F A182031 Euler transform of period 6 sequence [ 4, 8, 1, 8, 4, 2, ...].
%F A182031 A002513(3*n + 2) = 3 * a(n).
%e A182031 1 + 4*x + 18*x^2 + 53*x^3 + 163*x^4 + 414*x^5 + 1059*x^6 + 2431*x^7 + ...
%e A182031 q^5 + 4*q^13 + 18*q^21 + 53*q^29 + 163*q^37 + 414*q^45 + 1059*q^53 + ...
%t A182031 eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-5/8)*(eta[q^3]*eta[q^6])^3/(eta[q]*eta[q^2])^4, {q, 0, 100}], q] (* _G. C. Greubel_, Apr 16 2018 *)
%o A182031 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^2 + A))^4, n))}
%o A182031 (PARI) q='q+O('q^99); Vec((eta(q^3)*eta(q^6))^3/(eta(q)*eta(q^2))^4) \\ _Altug Alkan_, Apr 16 2018
%Y A182031 Cf. A002513.
%K A182031 nonn
%O A182031 0,2
%A A182031 _Michael Somos_, Apr 07 2012