A187154 Expansion of psi(x^4) / phi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.

1, 2, 4, 8, 15, 26, 44, 72, 114, 178, 272, 408, 605, 884, 1276, 1824, 2580, 3616, 5028, 6936, 9498, 12922, 17468, 23472, 31369, 41700, 55156, 72616, 95172, 124202, 161436, 209016, 269616, 346562, 443952, 566856, 721530, 915642, 1158608, 1461968, 1839789

Offset

0,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/2) * eta(q^2) * eta(q^8)^2 / (eta(q)^2 * eta(q^4)) in powers of q.

Euler transform of period 8 sequence [ 2, 1, 2, 2, 2, 1, 2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A093085.

Convolution inverse of A093085. Convolution square is A107035.

a(n) ~ exp(sqrt(n)*Pi)/(16*n^(3/4)). - Vaclav Kotesovec, Sep 10 2015

Example

1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 26*x^5 + 44*x^6 + 72*x^7 + 114*x^8 + ...
q + 2*q^3 + 4*q^5 + 8*q^7 + 15*q^9 + 26*q^11 + 44*q^13 + 72*q^15 + 114*q^17 + ...

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 + x^k)^2 * (1 + x^(2*k)) * (1 + x^(4*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)
a[n_]:= SeriesCoefficient[EllipticTheta[2, 0, q^2]/(2*Sqrt[q]* EllipticTheta[3, 0, -q]), {q, 0, n}]; Table[A187154[n], {n, 0, 50}] (* G. C. Greubel, Dec 04 2017 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))}

Crossrefs

Cf. A093085, A107035.

Sequence in context: A133551 A114226 A210063 * A179001 A222147 A003241

Adjacent sequences: A187151 A187152 A187153 * A187155 A187156 A187157

Keyword

nonn

Author

Michael Somos, Mar 08 2011

Status

approved