A275326 Triangle read by rows, T(n,k) = ceiling(A275325(n,k)/2) for n>=0 and 0<=k<=n.

1, 0, 1, 0, 1, 0, 3, 0, 2, 1, 0, 10, 5, 0, 5, 4, 1, 0, 35, 28, 7, 0, 14, 14, 6, 1, 0, 126, 126, 54, 9, 0, 42, 48, 27, 8, 1, 0, 462, 528, 297, 88, 11, 0, 132, 165, 110, 44, 10, 1, 0, 1716, 2145, 1430, 572, 130, 13, 0, 429, 572, 429, 208, 65, 12, 1

Offset: 0

Author

Peter Luschny, Aug 15 2016

Keywords

• nonn
• tabf

Comments

An extension of the Catalan triangle A128899.

Examples

			Triangle starts:
[ n] [k=0,1,2,...] [row sum]
[ 0] [1] 1
[ 1] [0, 1] 1
[ 2] [0, 1] 1
[ 3] [0, 3] 3
[ 4] [0, 2, 1] 3
[ 5] [0, 10, 5] 15
[ 6] [0, 5, 4, 1] 10
[ 7] [0, 35, 28, 7] 70
[ 8] [0, 14, 14, 6, 1] 35
[ 9] [0, 126, 126, 54,  9] 315
[10] [0, 42, 48, 27, 8, 1] 126
[11] [0, 462, 528, 297, 88, 11] 1386
[12] [0, 132, 165, 110, 44, 10, 1] 462
		

Links

• Peter Luschny, Orbitals

Crossrefs

Cf. A057977, A093178, A128899, A275324 (row sums), A275325.

Programs

Sage

# uses[orbital_factors]
# Function orbital_factors is in A275325.
def half_orbital_factors(n):
    F = orbital_factors(n)
    return [f//2 for f in F] if n >= 2 else F
for n in (0..12): print(half_orbital_factors(n))

Formula

T(n,k) = A275325(n,k)/2 for n>=2.

T(n,1) = A057977(n) for n>=1 (the extended Catalan numbers).

For odd n: T(n,1) = Sum_{k>=0} T(n+1,k).

Main diagonal: T(n, floor(n/2)) = A093178(n).