This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 16 13:12 EDT 2018. Contains 316263 sequences. (Running on oeis4.)