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

0,4

Comments

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

Links

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

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

Example

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

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 *)

Crossrefs

Sequence in context: A079222 A033184 A171567 * A271025 A134379 A108087

Adjacent sequences: A110485 A110486 A110487 * A110489 A110490 A110491

Keyword

easy,nonn,tabl

Author

Paul Barry, Jul 22 2005

Status

approved