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

1,2

Comments

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

Links

Table of n, a(n) for n=1..50.

Formula

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

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

Example

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;
  ...

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 *)

Crossrefs

Cf. A000108, A001263, A002457, A085374.

Sequence in context: A107783 A047887 A120986 * A128567 A345394 A217204

Adjacent sequences: A095798 A095799 A095800 * A095802 A095803 A095804

Keyword

nonn,easy,nice,tabl

Author

Gary W. Adamson, Jun 07 2004

Extensions

Edited and extended by David Wasserman, Sep 24 2004

Status

approved