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%I M3870 #90 Apr 22 2023 18:18:51
%S 1,0,1,1,5,16,65,260,1085,4600,19845,86725,383251,1709566,7687615,
%T 34812519,158614405,726612216,3344696501,15462729645,71763732545,
%U 334236300200,1561686608685,7318223046860,34386154568375,161970182441556,764676831501575,3617755131480841
%N Number of noncommutative SL(2,C)-invariants of degree n in 5 variables.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vaclav Kotesovec, <a href="/A007043/b007043.txt">Table of n, a(n) for n = 0..1000</a>
%H Gert Almkvist, Warren Dicks, and Edward Formanek, <a href="http://dx.doi.org/10.1016/0021-8693(85)90183-8">Hilbert series of fixed free algebras and noncommutative classical invariant theory</a>, J. Algebra 93 (1985), no. 1, 189-214.
%H G. Almkvist, <a href="/A007043/a007043.pdf">Letter to N. J. A. Sloane, Apr. 1992</a>.
%H Eliahu Cohen, Tobias Hansen, and Nissan Itzhaki, <a href="http://arxiv.org/abs/1511.06623">From Entanglement Witness to Generalized Catalan Numbers</a>, arXiv:1511.06623 [quant-ph], 2015.
%H Thomas Curtright, T. S. Van Kortryk, and Cosmas Zachos, <a href="https://hal.archives-ouvertes.fr/hal-01345527">Spin Multiplicities</a>, hal-01345527, 2016.
%H R. K. Guy, <a href="/A007042/a007042.pdf">Letter to N. J. A. Sloane, Aug. 1992</a>.
%H R. K. Guy, <a href="/A007042/a007042_1.pdf">Parker's permutation problem involves the Catalan numbers</a>, preprint, 1992. (Annotated scanned copy)
%F From _Paul Barry_, Oct 18 2007: (Start)
%F a(n) = Sum{k=0..n} Sum{j=0..k} C(n,k)*C(k,j)*(-3)^(k-j)*A000108(j);
%F a(n) = (1/(2*Pi))*Integral_{x=0..4} (1 - 3*x + x^2)^n*sqrt(x*(4 - x))/x dx. (End)
%F G.f.: F(G^(-1)(x)), where F(t) := (t^2 + 3*t + 1)/((t + 1)*(4*t + 1)^(1/2)) and G(t) := t/(t^2 + 3*t + 1). - _Mark van Hoeij_, Oct 30 2011
%F a(n) ~ 5^n/(8*sqrt(Pi)*n^(3/2)) * (1 - 15/(16*n) + O(1/n^2)). - _Thomas Curtright_, Jun 17 2016, updated Jul 26 2016
%F D-finite with recurrence: 2*n*(2*n + 1)*(3*n - 5)*a(n) = (n-1)*(3*n - 2)*(19*n - 20)*a(n-1) + 10*(n-1)*n*(3*n - 5)*a(n-2) - 25*(n-2)*(n-1)*(3*n - 2)*a(n-3). - _Vaclav Kotesovec_, Jun 24 2016
%F a(n) = (1/Pi)*Integral_{x=0..2*Pi} (sin(5*x)/sin(x))^n*(sin(x))^2. - _Thomas Curtright_, Jun 24 2016
%p F := (t^2+3*t+1)/((t+1)*(4*t+1)^(1/2)); G := t/(t^2+3*t+1); Ginv := RootOf(numer(G-x),t); ogf := series(eval(F,t=Ginv),x=0,20); # _Mark van Hoeij_, Oct 30 2011
%t CoefficientList[Series[Sqrt[2]/Sqrt[(1 - x)*((1 + 5*x) + Sqrt[(1 - 5*x)*(1 - x)])], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 24 2016, after Almkvist, Dicks and Formanek *)
%t a[n_]:= c[0, n, 2]-c[1, n, 2]; c[j_, n_, s_]:= Sum[(-1)^k*Binomial[n, k]*Binomial[j - (2*s + 1)*k + n + n*s - 1, j - (2*s + 1)*k + n*s], {k, 0, Min[n, Floor[(j + n*s)/(2*s + 1)]]}]; Table[a[n], {n, 0, 20}] (* Thomas Curtright, Jul 26 2016 *)
%Y Cf. A000108, A348210 (column k=2).
%K nonn
%O 0,5
%A _N. J. A. Sloane_ and _Mira Bernstein_
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