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A072279 Dimension of n-th graded section of a certain Lie algebra. 8
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 (list; graph; refs; listen; history; text; internal format)
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
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
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

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Last modified March 1 13:16 EST 2024. Contains 370433 sequences. (Running on oeis4.)