A260600 Expansion of x * psi(x^3) * psi(x^12) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

0, 1, 1, 2, 4, 6, 9, 14, 20, 29, 42, 58, 80, 111, 149, 200, 268, 353, 463, 606, 784, 1011, 1299, 1656, 2104, 2664, 3354, 4208, 5264, 6555, 8138, 10076, 12428, 15288, 18758, 22944, 27996, 34081, 41377, 50124, 60592, 73075, 87951, 105652, 126652, 151547, 181015

Offset

0,4

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-5/6) * eta(q^6)^2 * eta(q^24)^2 / (eta(q) * eta(q^3) * eta(q^12)) in powers of q.

Euler transform of period 24 sequence [1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, -1, ...].

-2 * a(n) = A260574(4*n + 3).

a(n) ~ exp(sqrt(2*n/3)*Pi) / (24*sqrt(2*n)). - Vaclav Kotesovec, Oct 14 2015

Example

G.f. = x + x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 14*x^7 + 20*x^8 + ...
G.f. = q^11 + q^17 + 2*q^23 + 4*q^29 + 6*q^35 + 9*q^41 + 14*q^47 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(3/2)] EllipticTheta[ 2, 0, x^6] / ( 4 x^(7/8) QPochhammer[ x]), {x, 0, n}];
nmax=60; CoefficientList[Series[x*Product[(1-x^(6*k)) * (1-x^(24*k)) * (1+x^(3*k)) * (1+x^(12*k)) / ((1-x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^6 + A)^2 * eta(x^24 + A)^2 / (eta(x + A) * eta(x^3 + A) * eta(x^12 + A)), n))};
(PARI) q='q+O('q^99); concat(0, Vec(eta(q^6)^2*eta(q^24)^2 / (eta(q)*eta(q^3)*eta(q^12)))) \\ Altug Alkan, Mar 18 2018

Crossrefs

Cf. A260574.

Sequence in context: A295341 A139135 A097197 * A119737 A038718 A378307

Adjacent sequences: A260597 A260598 A260599 * A260601 A260602 A260603

Keyword

nonn

Author

Michael Somos, Jul 29 2015

Status

approved