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%I #32 Aug 21 2017 05:24:08
%S 1,4,6,16,45,144,440,1440,4680,15600,52344,177840,608160,2095920,
%T 7262640,25300032,88517520,310927680,1095923400,3874804560,
%U 13737892896,48829153920,173949483240,620963048160,2220904271040,7956987570576,28553731537320,102617166646800
%N Dimension of n-th graded section of a certain Lie algebra.
%C Dimensions of Lie algebra associated to Yang-Lee algebra in the A. Connes and M. Dubois-Violette paper. - _Roger L. Bagula_, May 25 2007
%H Alois P. Heinz, <a href="/A072279/b072279.txt">Table of n, a(n) for n = 0..1000</a>
%H Latham Boyle, Paul J. Steinhardt, <a href="https://arxiv.org/abs/1608.08220">Self-Similar One-Dimensional Quasilattices</a>, arXiv preprint arXiv:1608.08220 [math-ph], 2016.
%H A. Connes and M. Dubois-Violette, <a href="http://arXiv.org/abs/math.QA/0206205">Yang-Mills Algebra</a>, arXiv:math/0206205 [math.QA], 2002.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F Product_{n=1..inf} 1/(1-x^n)^a(n) = 1/((1-x^2)*(1-4*x+x^2)).
%F 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.
%F a(n) ~ (2+sqrt(3))^n / n. - _Vaclav Kotesovec_, Sep 11 2014
%p 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
%t 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_ *)
%Y Inverse EULER transform of A072335 (with its initial 1 omitted).
%Y Cf. A072337.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jul 15 2002
%E Edited by _N. J. A. Sloane_, May 16 2008 at the suggestion of _R. J. Mathar_
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