A072279 Dimension of n-th graded section of a certain Lie algebra.

1, 4, 6, 16, 45, 144, 440, 1440, 4680, 15600, 52344, 177840, 608160, 2095920, 7262640, 25300032, 88517520, 310927680, 1095923400, 3874804560, 13737892896, 48829153920, 173949483240, 620963048160, 2220904271040, 7956987570576, 28553731537320, 102617166646800

Offset

0,2

Comments

Dimensions of Lie algebra associated to Yang-Lee algebra in the A. Connes and M. Dubois-Violette paper. - Roger L. Bagula, May 25 2007

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Latham Boyle, Paul J. Steinhardt, Self-Similar One-Dimensional Quasilattices, arXiv preprint arXiv:1608.08220 [math-ph], 2016.

A. Connes and M. Dubois-Violette, Yang-Mills Algebra, arXiv:math/0206205 [math.QA], 2002.

N. J. A. Sloane, Transforms

Formula

Product_{n=1..inf} 1/(1-x^n)^a(n) = 1/((1-x^2)*(1-4*x+x^2)).

a(n) = (1/n) * Sum_{k|n} moebius(n/k) (t1^k + t2^k), where t1, t2 are the roots of x^2-4x+1.

a(n) ~ (2+sqrt(3))^n / n. - Vaclav Kotesovec, Sep 11 2014

Maple

with(numtheory): f:= proc(n) option remember; `if`(n<1, `if`(n=0, 1, 0), 4*(f(n-1)-f(n-3)) +f(n-4)) end: c:= proc(n) option remember; local j; n*f(n) -add(c(j)*f(n-j), j=1..n-1) end: a:= proc(n) option remember; local d; `if`(n=0, 1, add(mobius(n/d)*c(d), d=divisors(n))/n) end: seq(a(n), n=0..27); # Alois P. Heinz, Sep 09 2008

Mathematica

f[n_] := f[n] = If[n < 1, If[n == 0, 1, 0],  f[n-4] + 4*(f[n-1] - f[n-3])]; c[n_] := c[n] = n*f[n] - Sum[c[j]*f[n-j],  {j, 1, n-1}]; a[n_] := a[n] = If[n == 0, 1,  Sum[c[d]*MoebiusMu[n/d],  {d, Divisors[n]}]/n]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Mar 14 2014, after Alois P. Heinz *)

Crossrefs

Inverse EULER transform of A072335 (with its initial 1 omitted).

Cf. A072337.

Sequence in context: A231998 A056421 A032295 * A038236 A223269 A264471

Adjacent sequences: A072276 A072277 A072278 * A072280 A072281 A072282

Keyword

nonn,easy

Author

N. J. A. Sloane, Jul 15 2002

Extensions

Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar

Status

approved