A137614 Triangle read by rows: A000012 * A047812 as infinite lower triangular matrices.

1, 2, 1, 3, 4, 1, 4, 9, 8, 1, 5, 18, 28, 12, 1, 6, 31, 76, 63, 19, 1, 7, 51, 176, 232, 131, 27, 1, 8, 79, 370, 693, 617, 248, 39, 1, 9, 119, 722, 1821, 2284, 1458, 450, 53, 1, 10, 173, 1337, 4338, 7243, 6553, 3211, 773, 74, 1

Offset: 0

Author

Gary W. Adamson, Jan 30 2008

Keywords

• nonn
• tabl

Comments

Row sums = A014138: (1, 3, 8, 22, 64, 196, 625, ...).

From Petros Hadjicostas, Jun 01 2020: (Start)

We prove the claim above. From Guy (1992, 1993), we know that A000108(n) = Sum_{k=0..n-1} A047812(k) (the row sums of Parker's triangle are Catalan numbers).

We then have Sum_{k=0..n-1} T(n,k) = Sum_{k=0..n-1} Sum_{s=k+1..n} A047812(s,k) = Sum_{s=1..n} Sum_{k=0..s-1} A047812(s,k) = Sum_{s=1..n} A000108(s) = A014138(n) because A014138 contains partial sums of the Catalan numbers. (End)

Examples

			Triangle T(n,k) (with rows n >= 1 and columns k = 0..n-1) begins:
  1;
  2,  1;
  3,  4,   1;
  4,  9,   8,   1;
  5, 18,  28,  12,   1;
  6, 31,  76,  63,  19,  1;
  7, 51, 176, 232, 131, 27, 1;
  ...
		

Links

• R. K. Guy, Parker's permutation problem involves the Catalan numbers, preprint, 1992. (Annotated scanned copy)
• R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.

Crossrefs

Cf. A000012, A014138, A047812.

Programs

PARI

A(n, k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
T(n,k) = sum(s=k+1, n, A(s,k));
vector(15, n, vector(n, k, T(n, k-1))) \\ Petros Hadjicostas, Jun 01 2020

Formula

T(n,k) = Sum_{s=k+1..n} A047812(s,k) for n >= 1 and 0 <= k <= n-1. - Petros Hadjicostas, Jun 01 2020