A186753 T(n, k) = Sum_{i=0, n} (Sum_{j=0, k} C(i+j,i) * C(n-i+j,n-i) * C(i+k-j,k-j) * C(n-i+k-j,n-i)).

1, 2, 2, 3, 8, 3, 4, 20, 20, 4, 5, 40, 76, 40, 5, 6, 70, 216, 216, 70, 6, 7, 112, 511, 832, 511, 112, 7, 8, 168, 1064, 2568, 2568, 1064, 168, 8, 9, 240, 2016, 6768, 10036, 6768, 2016, 240, 9, 10, 330, 3552, 15840, 32680, 32680, 15840, 3552, 330, 10, 11, 440

Offset

0,2

Comments

Second row or column is A007290.

Links

Table of n, a(n) for n=0..56.

L. Carlitz, Some Binomial Coefficient Identities, Fibonacci Quart. 4 (1966), 323-331

M. E. Cohen and H. S. Sun, Some extensions of the Brock-Carlitz identity, Proc. Amer. Math. Soc. 76 (1979), 178-185

M. E. Cohen and H. S. Sun, Further generalizations of the Brock-Carlitz identity, Journal of Mathematical Analysis and Applications, Volume 82, Issue 2, August 1981, Pages 346-360

Formula

T(n,k) = T(k, n).

T(n,k) - T(n-1, k) - T(n, k-1) = C(n+k,k)^2 : Carlitz-Brock identity (cf links).

Example

Table starts:
0: 1, 2, 3, 4, 5,
1: 2, 8, 20, 40, 70,
2: 3, 20, 76, 216, 511,
3: 4, 40, 216, 832, 2568,
4: 5, 70, 511, 2568, 10036

Prog

(PARI) h(m, n) = sum(i=0, m, sum(j=0, n, binomial(i+j, i)*binomial(m-i+j, m-i)*binomial(i+n-j, n-j)*binomial(m-i+n-j, m-i)))

Crossrefs

Sequence in context: A141611 A234357 A145596 * A135835 A177696 A134574

Adjacent sequences: A186750 A186751 A186752 * A186754 A186755 A186756

Keyword

nonn,tabl

Author

Michel Marcus, Dec 19 2012

Status

approved