A032442 Expansion of 1 / Product_{k >= 1} (1-q^k)^2*(1-q^(11k))^2.

1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 481, 754, 1169, 1780, 2685, 3996, 5894, 8600, 12450, 17860, 25442, 35964, 50519, 70490, 97800, 134892, 185099, 252664, 343280, 464200, 625033, 837998, 1119114, 1488720, 1973210, 2606028, 3430238, 4500224, 5885540

Offset

0,2

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 = 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 1 / (f(-x) * f(-x^11))^2 in powers of x where f() is a Ramanujan theta function. - Michael Somos, Apr 21 2015

Expansion of q / eta(q)^2 * eta(q^11)^2 in powers of q. - Michael Somos, Apr 21 2015

Euler transform of period 11 sequence [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, ...]. - Michael Somos, Apr 21 2015

Given g.f. A(x), then B(q) = A(q)/q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2 * (w^2 + 16*v^2) - v^2 * (v + 4*u) * (w + 4*u). - Michael Somos, Apr 21 2015

G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^-1 (t/i)^-2 f(t) where q = exp(2 Pi i t). - Michael Somos, Apr 21 2015

G.f.: (Product_{k > 0} (1 - x^k)^2 * (1 - x^(11*k)))^-2.

Convolution inverse of A006571. Convolution with A028610 is A128525. - Michael Somos, Apr 21 2015

a(n) ~ exp(4*Pi*sqrt(n/11)) / (sqrt(2) * 11^(1/4) * n^(7/4)). - Vaclav Kotesovec, Oct 13 2015

Example

G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 110*x^7 + ...
G.f. = 1/q + 2 + 5*q + 10*q^2 + 20*q^3 + 36*q^4 + 65*q^5 + 110*q^6 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^11])^-2, {x, 0, n}]; (* Michael Somos, Apr 21 2015 *)
nmax=60; CoefficientList[Series[Product[1/((1-x^k)^2 * (1-x^(11*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^-2, n))}; /* Michael Somos, Apr 21 2015 */

Crossrefs

Cf. A006571, A028610, A128525.

Sequence in context: A103929 A121597 A000712 * A327293 A327292 A327291

Adjacent sequences: A032439 A032440 A032441 * A032443 A032444 A032445

Keyword

nonn

Author

N. J. A. Sloane

Status

approved