A256637 - OEIS

%I #12 Mar 12 2021 22:24:47

%S 1,-1,0,-2,1,0,0,2,0,2,0,0,0,-4,0,-4,-1,0,0,6,0,8,0,0,1,-10,0,-12,1,0,

%T 0,16,0,18,0,0,-2,-24,0,-28,-1,0,0,36,0,40,0,0,2,-52,0,-58,2,0,0,74,0,

%U 84,0,0,-2,-104,0,-116,-3,0,0,144,0,160,0,0,3

%N Expansion of psi(-q) * phi(-q^3)^2 / (q * psi(-q^3)^3) in powers of q where phi(), psi() 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="/A256637/b256637.txt">Table of n, a(n) for n = -1..1000</a>

%H Johannes Blümlein, <a href="https://arxiv.org/abs/1808.08128">Iterative Non-iterative Integrals in Quantum Field Theory</a>, arXiv:1808.08128 [hep-th], 2018.

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

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

%F Euler transform of period 12 sequence [ -1, 0, -2, -1, -1, -2, -1, -1, -2, 0, -1, 0, ...].

%F a(3*n + 1) = a(4*n + 1) = 0. a(2*n) = - A139137(n). a(4*n - 1) = A256626(n).

%F a(6*n) = - A132002(n). a(6*n + 2) = -2 * A139135(n). a(12*n - 1) = A256636(n). a(12*n + 3) = A101195(n).

%e G.f. = 1/q - 1 - 2*q^2 + q^3 + 2*q^6 + 2*q^8 - 4*q^12 - 4*q^14 - q^15 + ...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]* eta[q^3]*eta[q^4]*eta[q^6]/(eta[q^2]*eta[q^12]^3), {q, 0, n}]; Table[a[n], {n,-1,100}] (* _G. C. Greubel_, Mar 14 2018 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^6 + A) / (eta(x^2 + A) * eta(x^12 + A)^3), n))};

%Y Cf. A101195, A139137, A256626, A256636.

%K sign

%O -1,4

%A _Michael Somos_, Apr 06 2015