A107035 Expansion of q * (psi(q^4) / phi(-q))^2 in powers of q where phi(), psi() are Ramanujan theta functions.

1, 4, 12, 32, 78, 176, 376, 768, 1509, 2872, 5316, 9600, 16966, 29408, 50088, 83968, 138738, 226196, 364284, 580032, 913824, 1425552, 2203368, 3376128, 5130999, 7738136, 11585208, 17225472, 25444278, 37350816, 54504160, 79085568

Offset: 1

Author

Michael Somos, May 09 2005

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = q + 4*q^2 + 12*q^3 + 32*q^4 + 78*q^5 + 176*q^6 + 376*q^7 + 768*q^8 + ...
		

References

• R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 375. Eqs. (20), (21), (24)

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000
• Kevin Acres and David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A014969, A131124, A131126.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (1/4) (EllipticTheta[ 2, 0, q^2] / EllipticTheta[ 4, 0, q])^2, {q, 0, n}]; (* Michael Somos, Jun 13 2012 *)
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 *)
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 *)
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 *)

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))};

Formula

Expansion of (eta(q^2) / eta(q^4))^2 * (eta(q^8) / eta(q))^4 in powers of q.

Expansion of Fricke tau_8(omega) / 16 in powers of q = exp(2 Pi i omega).

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).

Expansion of ((phi(q) / phi(-q))^2 - 1) / 8 in powers of q where phi() is a Ramanujan theta function.

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).

Euler transform of period 8 sequence [ 4, 2, 4, 4, 4, 2, 4, 0, ...].

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.

G.f: x * Product_{k>0} (1 + x^k)^4 * (1 + x^(2*k))^2 * (1 + x^(4*k))^4.

Convolution inverse of A131124. A131126(n) = 4 * a(n) unless n=0. A014969(n) = 8 * a(n) unless n=0.

a(n) ~ exp(sqrt(2*n)*Pi) / (64 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015