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.

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

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.

Michael 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 * A360566 A283451 A372970

Adjacent sequences: A260409 A260410 A260411 * A260413 A260414 A260415

Keyword

nonn

Author

Michael Somos, Jul 24 2015

Status

approved