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A187154 Expansion of psi(x^4) / phi(-x) in powers of x where phi(), psi() are Ramanujan theta functions. 4
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 (list; graph; refs; listen; history; text; internal format)
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

M. 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,changed

AUTHOR

Michael Somos, Mar 08 2011

STATUS

approved

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Last modified December 11 19:09 EST 2017. Contains 295919 sequences.