|
%I #35 May 31 2023 21:03:29
%S 1,4,12,32,78,176,376,768,1509,2872,5316,9600,16966,29408,50088,83968,
%T 138738,226196,364284,580032,913824,1425552,2203368,3376128,5130999,
%U 7738136,11585208,17225472,25444278,37350816,54504160,79085568
%N Expansion of q * (psi(q^4) / phi(-q))^2 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).
%D R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 375. Eqs. (20), (21), (24)
%H Seiichi Manyama, <a href="/A107035/b107035.txt">Table of n, a(n) for n = 1..1000</a>
%H Kevin Acres and 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 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) / eta(q^4))^2 * (eta(q^8) / eta(q))^4 in powers of q.
%F Expansion of Fricke tau_8(omega) / 16 in powers of q = exp(2 Pi i omega).
%F Expansion of elliptic (1/8) * (-1 + 1 / sqrt(1 - lambda(z)) = (1/8) * (-1 + 1 / k') in powers of the nome q = exp(Pi i z).
%F Expansion of ((phi(q) / phi(-q))^2 - 1) / 8 in powers of q where phi() is a Ramanujan theta function.
%F Elliptic j(z) = 256 * (x^4 + 8*x^3 + 20*x^2 + 16*x + 1)^3 / (x * (x + 4) * (x + 2)^2) where x = tau_8(z).
%F Euler transform of period 8 sequence [ 4, 2, 4, 4, 4, 2, 4, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v - u^2 + 4 * v^2 + 8 * u * v + 32 * u * v^2.
%F G.f: x * Product_{k>0} (1 + x^k)^4 * (1 + x^(2*k))^2 * (1 + x^(4*k))^4.
%F Convolution inverse of A131124. A131126(n) = 4 * a(n) unless n=0. A014969(n) = 8 * a(n) unless n=0.
%F a(n) ~ exp(sqrt(2*n)*Pi) / (64 * 2^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 10 2015
%e G.f. = q + 4*q^2 + 12*q^3 + 32*q^4 + 78*q^5 + 176*q^6 + 376*q^7 + 768*q^8 + ...
%t a[ n_] := SeriesCoefficient[ (1/4) (EllipticTheta[ 2, 0, q^2] / EllipticTheta[ 4, 0, q])^2, {q, 0, n}]; (* _Michael Somos_, Jun 13 2012 *)
%t a[ n_] := With[ {m = ModularLambda[ Log[q] / (Pi I)]}, SeriesCoefficient[ (1/8) (-1 + 1 / Sqrt[1 - m]), {q, 0, n}]]; (* _Michael Somos_, Jun 13 2012 *)
%t nmax = 50; CoefficientList[Series[Product[(1 + x^k)^4 * (1 + x^(2*k))^2 * (1 + x^(4*k))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 10 2015 *)
%t QP = QPochhammer; s = (QP[q^2]/QP[q^4])^2*(QP[q^8]/QP[q])^4 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 30 2015, adapted from PARI *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x^4 + A))^2 * (eta(x^8 + A) / eta(x + A))^4, n))};
%Y Cf. A014969, A131124, A131126.
%K nonn
%O 1,2
%A _Michael Somos_, May 09 2005
|
