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%I #14 Dec 20 2012 13:22:14
%S 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,
%T 511,112,7,8,168,1064,2568,2568,1064,168,8,9,240,2016,6768,10036,6768,
%U 2016,240,9,10,330,3552,15840,32680,32680,15840,3552,330,10,11,440
%N 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)).
%C Second row or column is A007290.
%H L. Carlitz, <a href="http://www.fq.math.ca/Scanned/4-4/carlitz.pdf">Some Binomial Coefficient Identities</a>, Fibonacci Quart. 4 (1966), 323-331
%H M. E. Cohen and H. S. Sun, <a href="http://dx.doi.org/10.1090/S0002-9939-1979-0534414-5">Some extensions of the Brock-Carlitz identity</a>, Proc. Amer. Math. Soc. 76 (1979), 178-185
%H M. E. Cohen and H. S. Sun, <a href="http://dx.doi.org/10.1016/0022-247X(81)90200-6">Further generalizations of the Brock-Carlitz identity</a>, Journal of Mathematical Analysis and Applications, Volume 82, Issue 2, August 1981, Pages 346-360
%F T(n,k) = T(k, n).
%F T(n,k) - T(n-1, k) - T(n, k-1) = C(n+k,k)^2 : Carlitz-Brock identity (cf links).
%e Table starts:
%e 0: 1, 2, 3, 4, 5,
%e 1: 2, 8, 20, 40, 70,
%e 2: 3, 20, 76, 216, 511,
%e 3: 4, 40, 216, 832, 2568,
%e 4: 5, 70, 511, 2568, 10036
%o (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)))
%K nonn,tabl
%O 0,2
%A _Michel Marcus_, Dec 19 2012
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