A033878 Triangular array associated with Schroeder numbers.

1, 1, 1, 1, 3, 2, 1, 5, 10, 6, 1, 7, 22, 38, 22, 1, 9, 38, 98, 158, 90, 1, 11, 58, 194, 450, 698, 394, 1, 13, 82, 334, 978, 2126, 3218, 1806, 1, 15, 110, 526, 1838, 4942, 10286, 15310, 8558, 1, 17, 142, 778, 3142, 9922, 25150, 50746, 74614, 41586

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy

Comments

Transpose of triangular array A132372. - Michel Marcus, May 02 2015

Examples

			This triangle reads:
1
1  1
1  3   2
1  5   10   6
1  7   22   38   22
1  9   38   98   158   90
1  11  58   194  450   698   394
1  13  82   334  978   2126  3218   1806
1  15  110  526  1838  4942  10286  15310   558
1  17  142  778  3142  9922  25150  50746  74614  41586
		

Links

• E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.

Crossrefs

Cf. A132372, A006318.

Programs

PARI

lgs(n) = if( n<1, 1, sum( k=0, n, 2^k * binomial( n, k) * binomial( n, k-1)) / n) /* A006318 */
T(n, k) = if (k>n, 0, if (k==0, 1, if (n==0, 1, if ((n==k), lgs(n-1), T(n,k-1) + T(n-1,k-1) + T(n-1,k)))));
tabl(nn) = {for (n=0, nn, for (m=0, n, print1(T(n, m), ", ");); print(););} \\ Michel Marcus, May 02 2015

Python

from functools import cache
@cache
def T(n: int, k: int) -> int:
    if n <  0: return 0
    if k == 0: return 1
    if n == k: return sum(T(n-1, k) for k in range(n))
    return T(n, k-1) + T(n-1, k-1) + T(n-1, k)
for n in range(10):
    print([T(n, k) for k in range(n+1)])  # Peter Luschny, Dec 26 2024

Extensions

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003