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A260412 Expansion of psi(x^2) * psi(x^3) / f(-x^2, -x^10) in powers of x where psi(), f(,) are Ramanujan theta functions. 2
1, 0, 2, 1, 2, 2, 3, 2, 3, 4, 4, 5, 7, 6, 9, 10, 11, 12, 13, 15, 17, 19, 21, 24, 28, 30, 35, 37, 41, 47, 52, 56, 62, 69, 75, 83, 92, 99, 110, 121, 131, 143, 157, 170, 186, 203, 219, 239, 260, 281, 307, 332, 359, 389, 421, 453, 491, 530, 570, 617, 665, 714, 770 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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 q^(1/24) * eta(q^4)^3 * eta(q^6)^3 / (eta(q^2)^2 * eta(q^3) * eta(q^12)^2) in powers of q.

Euler transform of period 12 sequence [ 0, 2, 1, -1, 0, 0, 0, -1, 1, 2, 0, -1, ...].

a(n) ~ exp(Pi*sqrt(n/6)) / (4*sqrt(n)). - Vaclav Kotesovec, Jul 11 2016

EXAMPLE

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

G.f. = 1/q + 2*q^47 + q^71 + 2*q^95 + 2*q^119 + 3*q^143 + 2*q^167 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (4 x^(5/8) QPochhammer[ x^2, x^12] QPochhammer[ x^10, x^12] QPochhammer[ x^12]), {x, 0, n}];

nmax = 50; CoefficientList[Series[Product[(1-x^(4*k))^3 * (1-x^(6*k))^3 / ((1-x^(2*k))^2 * (1-x^(3*k)) * (1-x^(12*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 11 2016 *)

PROG

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

CROSSREFS

Sequence in context: A289120 A025066 A060426 * A283451 A172245 A238781

Adjacent sequences:  A260409 A260410 A260411 * A260413 A260414 A260415

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 24 2015

STATUS

approved

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Last modified April 6 21:24 EDT 2020. Contains 333286 sequences. (Running on oeis4.)