A275326 - OEIS

%I #12 Mar 14 2020 10:32:49

%S 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,

%T 54,9,0,42,48,27,8,1,0,462,528,297,88,11,0,132,165,110,44,10,1,0,1716,

%U 2145,1430,572,130,13,0,429,572,429,208,65,12,1

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

%C An extension of the Catalan triangle A128899.

%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>

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

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

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

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

%e Triangle starts:

%e [ n] [k=0,1,2,...] [row sum]

%e [ 0] [1] 1

%e [ 1] [0, 1] 1

%e [ 2] [0, 1] 1

%e [ 3] [0, 3] 3

%e [ 4] [0, 2, 1] 3

%e [ 5] [0, 10, 5] 15

%e [ 6] [0, 5, 4, 1] 10

%e [ 7] [0, 35, 28, 7] 70

%e [ 8] [0, 14, 14, 6, 1] 35

%e [ 9] [0, 126, 126, 54,  9] 315

%e [10] [0, 42, 48, 27, 8, 1] 126

%e [11] [0, 462, 528, 297, 88, 11] 1386

%e [12] [0, 132, 165, 110, 44, 10, 1] 462

%o (Sage) # uses[orbital_factors]

%o # Function orbital_factors is in A275325.

%o def half_orbital_factors(n):

%o     F = orbital_factors(n)

%o     return [f//2 for f in F] if n >= 2 else F

%o for n in (0..12): print(half_orbital_factors(n))

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

%K nonn,tabf

%O 0,7

%A _Peter Luschny_, Aug 15 2016