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%I #20 Dec 13 2015 16:08:12
%S 1,2,1,5,6,1,14,30,12,1,42,140,100,20,1,132,630,700,250,30,1,429,2772,
%T 4410,2450,525,42,1,1430,12012,25872,20580,6860,980,56,1,4862,51480,
%U 144144,155232,74088,16464,1680,72,1,16796,218790,772200,1081080,698544
%N Square of Narayana triangle A001263: View A001263 as a lower triangular matrix. Then the square of that matrix is also lower triangular. Sequence gives this lower triangle, read by rows.
%C The first three columns are A000108 (the Catalan numbers), A002457 and A085374.
%F T(n, k) = Sum_{i = k..n} A001263(n, i)*A001263(i, k).
%F T(n, n-1) = n*(n-1).
%e The first 3 rows are 1; 2, 1; 5, 6, 1; since the first 3 rows of the Narayana triangle in matrix format are M = [1 0 0 / 1 1 0 / 1 3 1]. Then M^2 = [1 0 0 / 2 1 0 / 5 6 1].
%e Triangle starts:
%e 1;
%e 2, 1;
%e 5, 6, 1;
%e 14, 30, 12, 1;
%e 42, 140, 100, 20, 1;
%e ...
%t t[n_, k_] = Sum[1/(i*k)*(Binomial[i-1, k-1]*Binomial[i, k-1]* Binomial[n-1, i-1]*Binomial[n, i-1]), {i, k, n}];
%t Flatten[Table[t[n, k], {n, 1, 10}, {k, 1, n}]][[1;;50]] (* _Jean-François Alcover_, Jul 21 2011 *)
%Y Cf. A000108, A001263, A002457, A085374.
%K nonn,easy,nice,tabl
%O 1,2
%A _Gary W. Adamson_, Jun 07 2004
%E Edited and extended by _David Wasserman_, Sep 24 2004
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