A176850 a(n,k) is the number of ways to choose integers i,j from {0,1,...,k} such that the inequalities |i-j| <= n <= i+j are satisfied.

1, 2, 3, 1, 3, 6, 6, 3, 1, 4, 9, 11, 10, 6, 3, 1, 5, 12, 16, 17, 15, 10, 6, 3, 1, 6, 15, 21, 24, 24, 21, 15, 10, 6, 3, 1, 7, 18, 26, 31, 33, 32, 28, 21, 15, 10, 6, 3, 1, 8, 21, 31, 38, 42, 43, 41, 36, 28, 21, 15, 10, 6, 3, 1, 9, 24, 36, 45, 51, 54, 54, 51, 45, 36, 28, 21, 15, 10, 6, 3, 1, 10, 27, 41, 52, 60, 65, 67, 66, 62, 55, 45, 36, 28, 21, 15, 10, 6, 3

Offset: 0

Author

Sean Murray, Apr 27 2010

Keywords

• tabf
• nonn
• easy

Comments

The rows are of length 1, 3, 5, 7, ...

a(n,k) is also the number of independent rank n tensor operators to appear in the tensor product of two spaces each spanned by k+1 tensor operators of ranks 0 to k,

{Y_{l,m},l=0,1,...,k, m:-l,-l+1,...,l} times {Y'_{l'm'}, l'=0,1,...,k, m':-l,-l+1,...,l}.

Basis elements of the tensor product space are given by

psi^{l,l'}{p,q} = Sum{m,m'} C^{ll'p}{mm'q} Y{l,m}Y'_{l'm'}

for all l,l' = 0,1,...,k and where p = |l-l'|, |l-l'|+1, ..., l+l' is the rank, q=-p, -p+1,...,p and where C^{ll'p}_{mm'q} are the Clebsch-Gordon coefficients.

Sum_{k=0..2*n+1} a(n,k)*(2*k+1) = (n+1)^4. - L. Edson Jeffery, Oct 29 2012

Sum_{k=0..2*n+1} (a(n,k) - a(n-1,k))*(2*k+1) = n^4 - (n-1)^4 = A005917(n+1), for n > 0. - L. Edson Jeffery, Nov 02 2012

Examples

			Triangle begins
   1;
   2,  3,  1;
   3,  6,  6,  3,  1;
   4,  9, 11, 10,  6,  3,  1;
   5, 12, 16, 17, 15, 10,  6,  3,  1;
   6, 15, 21, 24, 24, 21, 15, 10,  6,  3,  1;
   7, 18, 26, 31, 33, 32, 28, 21, 15, 10,  6,  3,  1;
   8, 21, 31, 38, 42, 43, 41, 36, 28, 21, 15, 10,  6,  3,  1;
   9, 24, 36, 45, 51, 54, 54, 51, 45, 36, 28, 21, 15, 10,  6,  3,  1;
  10, 27, 41, 52, 60, 65, 67, 66, 62, 55, 45, 36, 28, 21, 15, 10,  6,  3,  1;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10200 (rows n = 0..100, flattened)
• Eliahu Cohen, Tobias Hansen, and Nissan Itzhaki, From Entanglement Witness to Generalized Catalan Numbers, arXiv:1511.06623 [quant-ph], 2015.

Crossrefs

Cf. A005917.

Programs

Maple

Seq:=[]: for k from 0 to 15 do for n from 0 to k do Seq:= [op(Seq), -(3/2)*n^2+2*k*n+(1/2)*n+k+1] end do; for n from k+1 to 2*k do Seq:= [op(Seq), (1/2)*(2*k-n+1)*(2*k-n+2)] end do; end do; Seq;

Mathematica

Table[If[n <= k, -(3/2)*n^2 + 2*k*n + n/2 + k + 1, (2*k - n + 1)*(2*k - n + 2)/2], {k, 0, 8}, {n, 0, 2 k}] // Flatten (* Michael De Vlieger, Jul 10 2022 *)

Formula

a(n,k) = -(3/2)*n^2 + 2*k*n + n/2 + k + 1 for n=0,1,...,k, a(n) = (2*k-n+1)*(2*k-n+2)/2 for n = k+1,...,2*k.

Extensions

Edited by Sean Murray, Oct 05 2011