A123631 Expansion of q / (chi(-q) * chi(-q^11))^2 in powers of q where chi() is a Ramanujan theta function.

1, 2, 3, 6, 9, 14, 22, 32, 46, 66, 93, 130, 180, 244, 331, 444, 590, 780, 1024, 1334, 1730, 2234, 2867, 3664, 4660, 5904, 7449, 9364, 11728, 14638, 18211, 22584, 27927, 34436, 42342, 51924, 63523, 77512, 94364, 114624, 138920, 168012, 202786, 244270

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 22 sequence [ 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, ...].

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

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u^2 + v^2)^2 - u*v * (1 + 3*(u+v) + 4*u*v)^2.

a(n) ~ exp(2*Pi*sqrt(2*n/11)) / (2^(11/4) * 11^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2017

Example

G.f. = q + 2*q^2 + 3*q^3 + 6*q^4 + 9*q^5 + 14*q^6 + 22*q^7 + 32*q^8 + 46*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q / (QPochhammer[ q, q^2] QPochhammer[ q^11, q^22])^2, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ q / (Product[ 1 - q^k, {k, 1, n, 2}] Product[ 1 - q^k, {k, 11, n, 22}])^2, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
nmax = 60; CoefficientList[Series[Product[((1+x^k)*(1+x^(11*k)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2017 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^22 + A) / (eta(x + A) * eta(x^11 + A)))^2, n))};

Crossrefs

Sequence in context: A128518 A022567 A134004 * A228364 A018060 A115856

Adjacent sequences: A123628 A123629 A123630 * A123632 A123633 A123634

Keyword

nonn

Author

Michael Somos, Oct 03 2006

Status

approved