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A252501 Triangle T read by rows: T(n,k) = binomial(2*n+1,k)*binomial(n,k), n>=0, 0<=k<=n.

%I A252501 #9 Dec 18 2014 04:07:14
%S A252501 1,1,3,1,10,10,1,21,63,35,1,36,216,336,126,1,55,550,1650,1650,462,1,
%T A252501 78,1170,5720,10725,7722,1716,1,105,2205,15925,47775,63063,35035,6435,
%U A252501 1,136,3808,38080,166600,346528,346528,155584,24310
%N A252501 Triangle T read by rows: T(n,k) = binomial(2*n+1,k)*binomial(n,k), n>=0, 0<=k<=n.
%H A252501 Marc Chamberland and Karl Dilcher, <a href="http://dx.doi.org/10.1016/j.jnt.2009.05.010">A Binomial Sum Related to Wolstenholme's Theorem</a>, J. Number Theory, Vol. 171, Issue 11 (Nov. 2009), pp. 2659-2672 (see Thm. 2.3).
%e A252501 Triangle T begins:
%e A252501 .1
%e A252501 .1.....3
%e A252501 .1....10.....10
%e A252501 .1....21.....63......35
%e A252501 .1....36....216.....336......126
%e A252501 .1....55....550....1650.....1650......462
%e A252501 .1....78...1170....5720....10725.....7722.....1716
%e A252501 .1...105...2205...15925....47775....63063....35035.....6435
%e A252501 .1...136...3808...38080...166600...346528...346528...155584...24310
%t A252501 Flatten[Table[Binomial[2*n + 1, k]*Binomial[n, k], {n, 0, 8}, {k, 0, n}]] (* Replace Flatten[] with Grid[] to get the triangle. *)
%Y A252501 Cf. A000012 (col. 1), A014105 (col. 2),  A001700 (diag), A045721 (row sums).
%Y A252501 Cf. A234839, A252355.
%K A252501 nonn,tabl
%O A252501 0,3
%A A252501 _L. Edson Jeffery_, Dec 17 2014