A007043 Number of noncommutative SL(2,C)-invariants of degree n in 5 variables.

1, 0, 1, 1, 5, 16, 65, 260, 1085, 4600, 19845, 86725, 383251, 1709566, 7687615, 34812519, 158614405, 726612216, 3344696501, 15462729645, 71763732545, 334236300200, 1561686608685, 7318223046860, 34386154568375, 161970182441556, 764676831501575, 3617755131480841

Offset: 0

Author

N. J. A. Sloane and Mira Bernstein

Keywords

• nonn

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• Gert Almkvist, Warren Dicks, and Edward Formanek, Hilbert series of fixed free algebras and noncommutative classical invariant theory, J. Algebra 93 (1985), no. 1, 189-214.
• G. Almkvist, Letter to N. J. A. Sloane, Apr. 1992.
• Eliahu Cohen, Tobias Hansen, and Nissan Itzhaki, From Entanglement Witness to Generalized Catalan Numbers, arXiv:1511.06623 [quant-ph], 2015.
• Thomas Curtright, T. S. Van Kortryk, and Cosmas Zachos, Spin Multiplicities, hal-01345527, 2016.
• R. K. Guy, Letter to N. J. A. Sloane, Aug. 1992.
• R. K. Guy, Parker's permutation problem involves the Catalan numbers, preprint, 1992. (Annotated scanned copy)

Crossrefs

Cf. A000108, A348210 (column k=2).

Programs

Maple

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

Mathematica

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 *)
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 *)

Formula

From Paul Barry, Oct 18 2007: (Start)

a(n) = Sum{k=0..n} Sum{j=0..k} C(n,k)*C(k,j)*(-3)^(k-j)*A000108(j);

a(n) = (1/(2*Pi))*Integral_{x=0..4} (1 - 3*x + x^2)^n*sqrt(x*(4 - x))/x dx. (End)

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

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

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

a(n) = (1/Pi)*Integral_{x=0..2*Pi} (sin(5*x)/sin(x))^n*(sin(x))^2. - Thomas Curtright, Jun 24 2016