A260414 Expansion of psi(x^3) * psi(x^6) / f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions.

1, 0, 0, 1, 1, 0, 1, 1, 2, 2, 1, 2, 3, 2, 2, 4, 5, 4, 5, 6, 7, 7, 7, 9, 12, 11, 11, 15, 16, 16, 18, 21, 24, 25, 26, 31, 36, 35, 38, 45, 50, 51, 55, 63, 69, 73, 77, 87, 98, 101, 107, 122, 132, 138, 149, 164, 180, 190, 201, 222, 243, 254, 271, 300, 324, 340, 364

Offset

0,9

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^(-23/24) * eta(q^6) * eta(q^12)^2 / (eta(q^3) * eta(q^4)) in powers of q.

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

a(n) ~ exp(Pi*sqrt(n/6)) / (12*sqrt(n)). - Vaclav Kotesovec, Jul 11 2016

Example

G.f. = 1 + x^3 + x^4 + x^6 + x^7 + 2*x^8 + 2*x^9 + x^10 + 2*x^11 + ...
G.f. = q^23 + q^95 + q^119 + q^167 + q^191 + 2*q^215 + 2*q^239 + q^263 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(3/2)] EllipticTheta[ 2, 0, x^3] / (4 x^(9/8) QPochhammer[ x^4]), {x, 0, n}];
nmax = 100; CoefficientList[Series[Product[(1+x^(3*k)) * (1-x^(12*k))^2 / (1-x^(4*k)), {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^6 + A) * eta(x^12 + A)^2 / (eta(x^3 + A) * eta(x^4 + A)), n))};

Crossrefs

Sequence in context: A294186 A294185 A035462 * A160735 A216338 A227725

Adjacent sequences: A260411 A260412 A260413 * A260415 A260416 A260417

Keyword

nonn

Author

Michael Somos, Jul 24 2015

Status

approved