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A259529 Expansion of psi(-x^3)^2 / psi(-x) in powers of x where psi() is a Ramanujan theta function. 3
1, 1, 1, 0, 1, 2, 2, 2, 3, 3, 3, 3, 5, 6, 5, 6, 8, 9, 10, 10, 13, 15, 15, 17, 20, 23, 24, 25, 30, 34, 36, 39, 45, 50, 53, 57, 65, 73, 77, 83, 94, 104, 110, 118, 132, 145, 154, 166, 185, 201, 214, 230, 253, 276, 293, 316, 346, 375, 399, 427, 467, 505, 537, 575 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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..2500

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of f(x, x^5)^2 / f(x) in powers of x where f(,) is the general Ramanujan theta function.

Expansion of q^(-5/8) * eta(q^2) * eta(q^3)^2 * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^6)^2) in powers of q.

Euler transform of period 12 sequence [ 1, 0, -1, 1, 1, 0, 1, 1, -1, 0, 1, -1, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (768 t)) = (16/3)^1/2 (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A259538.

G.f.: Product_{k>0} (1 + x^k + x^(2*k)) * (1 - x^(3*k)) * (1 - x^(2*k) + x^(4*k)) * (1 + x^(6*k)).

a(n) ~ exp(Pi*sqrt(n/6)) / (6*sqrt(n)). - Vaclav Kotesovec, Jul 11 2016

EXAMPLE

G.f. = 1 + x + x^2 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + ...

G.f. = q^5 + q^13 + q^21 + q^37 + 2*q^45 + 2*q^53 + 2*q^61 + 3*q^69 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -1, 0, 1, -1, -1, 0, -1, -1, 1, 0, -1, 1}[[Mod[k, 12, 1]]], {k, n}], {x, 0, n}];

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3, x^6] QPochhammer[ x^12])^2 / ( QPochhammer[ x, x^2] QPochhammer[ x^4]), {x, 0, n}];

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, -1, 0, 1, -1, -1, 0, -1, -1, 1, 0, -1][k%12 + 1]), n))};

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2), n))};

CROSSREFS

Cf. A259538.

Sequence in context: A235130 A131410 A202453 * A196052 A080773 A134598

Adjacent sequences:  A259526 A259527 A259528 * A259530 A259531 A259532

KEYWORD

nonn

AUTHOR

Michael Somos, Jun 29 2015

STATUS

approved

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Last modified May 31 16:29 EDT 2020. Contains 334748 sequences. (Running on oeis4.)