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A190909
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Triangle read by rows: T(n,k) = binomial(n+k,n-k) * k! / floor(k/2)!^2.
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3
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1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 30, 42, 6, 1, 15, 70, 168, 54, 30, 1, 21, 140, 504, 270, 330, 20, 1, 28, 252, 1260, 990, 1980, 260, 140, 1, 36, 420, 2772, 2970, 8580, 1820, 2100, 70, 1, 45, 660, 5544, 7722, 30030, 9100, 16800, 1190, 630
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OFFSET
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0,5
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COMMENTS
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The triangle may be regarded as a generalization of the triangle A063007.
A063007(n,k) = binomial(n+k, n-k)*(2*k)$;
T(n,k) = binomial(n+k, n-k)*(k)$.
Here n$ denotes the swinging factorial A056040(n). As A063007 is a decomposition of the central Delannoy numbers A001850, a combinatorial interpretation of T(n,k) in terms of lattice paths can be expected.
T(n,n) = A056040(n) which can be seen as extended central binomial numbers.
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LINKS
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FORMULA
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EXAMPLE
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[0] 1
[1] 1, 1
[2] 1, 3, 2
[3] 1, 6, 10, 6
[4] 1, 10, 30, 42, 6
[5] 1, 15, 70, 168, 54, 30
[6] 1, 21, 140, 504, 270, 330, 20
[7] 1, 28, 252, 1260, 990, 1980, 260, 140
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MAPLE
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A190909 := (n, k) -> binomial(n+k, n-k)*k!/iquo(k, 2)!^2:
seq(print(seq(A190909(n, k), k=0..n)), n=0..7);
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MATHEMATICA
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Flatten[Table[Binomial[n+k, n-k] k!/(Floor[k/2]!)^2, {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Mar 25 2012 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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