A214868 Triangle T read by rows: T(n,0) = T(n,n) = 1 for n>=0, for n>=2 and 1<=k<=n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) if k = [n/2] or k = [(n+1)/2], else T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 5, 1, 1, 7, 11, 11, 7, 1, 1, 9, 23, 22, 23, 9, 1, 1, 11, 39, 45, 45, 39, 11, 1, 1, 13, 59, 107, 90, 107, 59, 13, 1, 1, 15, 83, 205, 197, 197, 205, 83, 15, 1

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Formula

Sum_{k, 0<=k<=n} T(n,k) = A110110(n), number of symmetric Schroeder paths of length 2n.

Sum_{k, 0<=k<=n-2} T(n+k,k) = A065096(n-1), n>=2.

T(2n,n) = A006318(n), large Schroeder numbers.

T(2n+1,n) = A001003(n+1), little Schroeder numbers.

T(n,0) = A000012(n).

T(n,1) = A004280(n).

T(n+2,2) = A142463(n) = A132209(n), n>0.

Example

Triangle begins
1
1, 1
1, 2, 1
1, 3, 3, 1
1, 5, 6, 5, 1
1, 7, 11, 11, 7, 1
1, 9, 23, 22, 23, 9, 1
1, 11, 39, 45, 45, 39, 11, 1
1, 13, 59, 107, 90, 107, 59, 13, 1
1, 15, 83, 205, 197, 197, 205, 83, 15, 1
1, 17, 111, 347, 509, 394, 509, 347, 111, 17, 1
1, 19, 143, 541, 1061, 903, 903, 1061, 541, 143, 19, 1
1, 21, 179, 795, 1949, 2473, 1806, 2473, 1949, 795, 179, 21, 1
...

Crossrefs

Cf. A001003, A006318, A065096, A110110

Sequence in context: A230859 A213086 A050446 * A144048 A292193 A258708

Adjacent sequences: A214865 A214866 A214867 * A214869 A214870 A214871

Keyword

nonn,tabl

Author

Philippe Deléham, Mar 10 2013

Status

approved