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A176850 a(n,k) is the number of ways to choose integers i,j from {0,1,...,k} such that the inequality |i-j|<= n <= i+j is satisfied. 1
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 (list; graph; refs; listen; history; text; internal format)
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 of 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

Table of n, a(n) for n=0..98.

Eliahu Cohen, Tobias Hansen, 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:

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;

CROSSREFS

Cf. A005917.

Sequence in context: A101912 A208522 A209569 * 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

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Last modified February 20 02:07 EST 2018. Contains 299357 sequences. (Running on oeis4.)