A095801 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.

1, 2, 1, 5, 6, 1, 14, 30, 12, 1, 42, 140, 100, 20, 1, 132, 630, 700, 250, 30, 1, 429, 2772, 4410, 2450, 525, 42, 1, 1430, 12012, 25872, 20580, 6860, 980, 56, 1, 4862, 51480, 144144, 155232, 74088, 16464, 1680, 72, 1, 16796, 218790, 772200, 1081080, 698544

Offset: 1

Author

Gary W. Adamson, Jun 07 2004

Keywords

• nonn
• easy
• nice
• tabl

Comments

The first three columns are A000108 (the Catalan numbers), A002457 and A085374.

Examples

			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].
Triangle starts:
   1;
   2,   1;
   5,   6,   1;
  14,  30,  12,  1;
  42, 140, 100, 20, 1;
  ...
		

Crossrefs

Cf. A000108, A001263, A002457, A085374.

Programs

Mathematica

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}];
Flatten[Table[t[n, k], {n, 1, 10}, {k, 1, n}]][[1;;50]] (* Jean-François Alcover, Jul 21 2011 *)

Formula

T(n, k) = Sum_{i = k..n} A001263(n, i)*A001263(i, k).

T(n, n-1) = n*(n-1).

Extensions

Edited and extended by David Wasserman, Sep 24 2004