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

Author

N. J. A. Sloane, Jul 15 2002

Keywords

• nonn
• easy

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

Crossrefs

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

Cf. A072337.

Programs

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 *)

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

Extensions

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