A128518 Expansion of q^(-1) * (chi(-q^13) / chi(-q))^2 in powers of q where chi() is a Ramanujan theta function.

1, 2, 3, 6, 9, 14, 22, 32, 46, 66, 93, 128, 176, 236, 315, 420, 550, 718, 932, 1198, 1534, 1956, 2476, 3120, 3919, 4896, 6095, 7562, 9341, 11504, 14126, 17284, 21090, 25666, 31140, 37692, 45515, 54818, 65878, 79000, 94523, 112872, 134522, 160004

Offset

-1,2

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 = -1..1000

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 (eta(q^2) * eta(q^13) / (eta(q) * eta(q^26)))^2 in powers of q.

Euler transform of period 26 sequence [ 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 - v) * (w^2 - v) - u*w * (4*(1+v^2) - 4*v).

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u*v - u - v)^3 - u*v * (u+v - 1) * (u^2 + v^2 + 1).

G.f. is a period 1 Fourier series which satisfies f(-1 / (26 t)) = f(t) where q = exp(2 Pi i t).

G.f.: (1/x)* (Product_{k>0} P(x^k))^-2 where P(x) is the 26th cyclotomic polynomial of degree 12.

a(n) = A058597(n) unless n = 0.

a(n) ~ exp(2*Pi*sqrt(2*n/13)) / (2^(3/4) * 13^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015

Example

G.f. = 1/q + 2 + 3*q + 6*q^2 + 9*q^3 + 14*q^4 + 22*q^5 + 32*q^6 + 46*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^2] QPochhammer[ q^13] / (QPochhammer[ q] QPochhammer[ q^26]))^2, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
nmax=60; CoefficientList[Series[Product[((1+x^k) / (1+x^(13*k)))^2, {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^2 + A) * eta(x^13 + A) / (eta(x + A) * eta(x^26 + A)))^2, n))};
(Magma) A := Basis( CuspForms( Gamma0(26), 2), 46);  B<q> := A[1] / A[2]; B; /* Michael Somos, Nov 30 2014 */

Crossrefs

Cf. A058597.

Sequence in context: A308251 A058609 A333697 * A022567 A134004 A123631

Adjacent sequences: A128515 A128516 A128517 * A128519 A128520 A128521

Keyword

nonn

Author

Michael Somos, Mar 06 2007

Status

approved