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A275326
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Triangle read by rows, T(n,k) = ceiling(A275325(n,k)/2) for n>=0 and 0<=k<=n.
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1
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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
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OFFSET
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0,7
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COMMENTS
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An extension of the Catalan triangle A128899.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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