A260515 Expansion of phi(x^2) * chi(x)^4 in powers of x where phi(), chi() are Ramanujan theta functions.

1, 4, 8, 16, 29, 44, 72, 112, 162, 244, 352, 496, 703, 968, 1320, 1792, 2405, 3204, 4240, 5568, 7259, 9416, 12144, 15568, 19875, 25260, 31944, 40256, 50523, 63140, 78672, 97680, 120870, 149148, 183480, 225056, 275350, 335984, 408920, 496544, 601514, 727044

Offset

0,2

Comments

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 6^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A260514.

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^(1/6) * eta(q^2)^6 * eta(q^4) / (eta(q)^4 * eta(q^8)^2) in powers of q.

Euler transform of period 8 sequence [4, -2, 4, -3, 4, -2, 4, -1, ...].

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

Example

G.f. = 1 + 4*x + 8*x^2 + 16*x^3 + 29*x^4 + 44*x^5 + 72*x^6 + 112*x^7 + ...
G.f. = 1/q + 4*q^5 + 8*q^11 + 16*q^17 + 29*q^23 + 44*q^29 + 72*q^35 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2] QPochhammer[ -x, x^2]^4, {x, 0, n}];
nmax=60; CoefficientList[Series[Product[(1-x^k) * (1+x^k)^5 / ((1+x^(2*k)) * (1+x^(4*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^4 + A) / (eta(x + A)^4 * eta(x^8 + A)^2), n))};
(PARI) q='q+O('q^99); Vec(eta(q^2)^6*eta(q^4) / (eta(q)^4*eta(q^8)^2)) \\ Altug Alkan, Mar 18 2018

Crossrefs

Cf. A260514.

Sequence in context: A271649 A128441 A009861 * A301148 A302508 A030119

Adjacent sequences: A260512 A260513 A260514 * A260516 A260517 A260518

Keyword

nonn,changed

Author

Michael Somos, Jul 27 2015

Status

approved