A007043 Number of noncommutative SL(2,C)-invariants of degree n in 5 variables. (Formerly M3870)

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

0,5

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)

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

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 *)

Crossrefs

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

Sequence in context: A301958 A349568 A026525 * A128242 A323934 A166932

Adjacent sequences: A007040 A007041 A007042 * A007044 A007045 A007046

Keyword

nonn

Author

N. J. A. Sloane and Mira Bernstein

Status

approved