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

0,2

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

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.

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.

Example

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;

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 *)

Crossrefs

Cf. A005917.

Sequence in context: A208522 A209569 A368158 * A208516 A111808 A247046

Adjacent sequences: A176847 A176848 A176849 * A176851 A176852 A176853

Keyword

tabf,nonn,easy

Author

Sean Murray, Apr 27 2010

Extensions

Edited by Sean Murray, Oct 05 2011

Status

approved