A262984 Expansion of f(-x^2, -x^10) / f(-x, -x) in powers of x where f(, ) is Ramanujan's general theta function.

1, 2, 3, 6, 10, 16, 26, 40, 60, 90, 131, 188, 268, 376, 522, 720, 983, 1330, 1790, 2390, 3170, 4184, 5488, 7160, 9300, 12020, 15466, 19822, 25300, 32168, 40760, 51464, 64763, 81250, 101620, 126726, 157604, 195472, 241810, 298400, 367340, 451156, 552867, 676030

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 6, 7th equation.

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 psi(x^6) * phi(-x^2) / f(-x)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

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

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

-2 * a(n) = A262967(3*n + 2).

a(n) ~ 5^(1/4) * exp(sqrt(5*n/6)*Pi) / (2^(13/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2015

G.f.: Sum_{k>=0} x^k * (Product_{i=1..k} 1 + x^(2*i)) / Product_{i=1..2*k+1} 1 - x^i). [Ramanujan] - Michael Somos, Nov 18 2015

Example

G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 26*x^6 + 40*x^7 + ...
G.f. = q^2 + 2*q^5 + 3*q^8 + 6*q^11 + 10*q^14 + 16*q^17 + 26*q^20 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^3] EllipticTheta[ 4, 0, x^2] / (2 x^(3/4) QPochhammer[ x]^2), {x, 0, n}];
nmax=60; CoefficientList[Series[Product[(1-x^(12*k)) * (1+x^(6*k)) * (1+x^(2*k-1)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q^2)^2*eta(q^12)^2/(eta(q)^2*eta(q^4)*eta(q^6))) \\ Altug Alkan, Mar 19 2018

Crossrefs

Cf. A262967.

Sequence in context: A280908 A146163 A101277 * A201077 A355383 A023655

Adjacent sequences: A262981 A262982 A262983 * A262985 A262986 A262987

Keyword

nonn

Author

Michael Somos, Oct 06 2015

Status

approved