

A176850


a(n,k) is the number of ways to choose integers i,j from {0,1,...,k} such that the inequality ij<= 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
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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=ll',ll'+1,...,l+l' is the rank, q=p,p+1,...,p and where C^{ll'p}_{mm'q} are the ClebschGordon 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(n1,k))*(2*k+1) = n^4(n1)^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 [quantph], 2015.


FORMULA

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



