A110488 A number triangle based on the Catalan numbers.

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 14, 14, 10, 4, 1, 42, 42, 35, 17, 5, 1, 132, 132, 126, 74, 26, 6, 1, 429, 429, 462, 326, 137, 37, 7, 1, 1430, 1430, 1716, 1446, 726, 230, 50, 8, 1, 4862, 4862, 6435, 6441, 3858, 1434, 359, 65, 9, 1, 16796, 16796, 24310, 28770, 20532, 8952, 2582, 530, 82, 10, 1

Offset: 0

Author

Paul Barry, Jul 22 2005

Keywords

• easy
• nonn
• tabl

Comments

Columns include A000108, A001700, A049027(n+1), A076025(n+1). Rows sums are A110489, diagonal sums are A110490.

Examples

			Rows begin
   1;
   1,  1;
   2,  2,  1;
   5,  5,  3,  1;
  14, 14, 10,  4,  1;
  42, 42, 35, 17,  5,  1;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Programs

Mathematica

T[n_, 0] := CatalanNumber[n]; T[n_, 1] := CatalanNumber[n]; T[n_, n_] := 1; T[n_, k_] := Sum[2*(j + 1)*(k - 1)^j*Binomial[2 (n - k) + 1, n - k - j]/(n - k + j + 2), {j, 0, n - k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 28 2017 *)

Formula

T(n, k) = Sum_{j=0..(n-k)} 2*(j+1)*(k-1)^j*C(2*(n-k)+1, n-k-j)/(n-k+j+2).

Column k has g.f. x^k*c(x)/(1-k*x*c(x)) where c(x) is the g.f. of A000108.

T(n,0) = Catalan(n), T(n,1) = Catalan(n), T(n,n) = 1. - G. C. Greubel, Aug 28 2017