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

1,2

Comments

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

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

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

Example

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

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 *)

Prog

(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))};

Crossrefs

Cf. A014969, A131124, A131126.

Sequence in context: A084566 A208903 A079769 * A260145 A260778 A118885

Adjacent sequences: A107032 A107033 A107034 * A107036 A107037 A107038

Keyword

nonn

Author

Michael Somos, May 09 2005

Status

approved