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A155856
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Triangle T(n,k) = binomial(2*n-k, k)*(n-k)!, read by rows.
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6
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1, 1, 1, 2, 3, 1, 6, 10, 6, 1, 24, 42, 30, 10, 1, 120, 216, 168, 70, 15, 1, 720, 1320, 1080, 504, 140, 21, 1, 5040, 9360, 7920, 3960, 1260, 252, 28, 1, 40320, 75600, 65520, 34320, 11880, 2772, 420, 36, 1, 362880, 685440, 604800, 327600, 120120, 30888, 5544, 660, 45, 1
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OFFSET
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0,4
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COMMENTS
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Downward diagonals T(n+j, n) = j!*binomial(n+j, n) = j!*seq(j), where seq(j) are sequences A010965, A010967, ..., A011101, A017714, A017716, ..., A017764, for 6 <= j <= 50, respectively. - G. C. Greubel, Jun 04 2021
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LINKS
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FORMULA
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T(n,k) = binomial(2*n-k, k)*(n-k)!.
Sum_{k=0..floor(n/2)} T(n-k, k) = A155858(n) (diagonal sums).
G.f.: 1/(1-xy-x/(1-xy-x/(1-xy-2x/(1-xy-2x/(1-xy-3x/(1-.... (continued fraction).
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EXAMPLE
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Triangle begins:
1;
1, 1;
2, 3, 1;
6, 10, 6, 1;
24, 42, 30, 10, 1;
120, 216, 168, 70, 15, 1;
720, 1320, 1080, 504, 140, 21, 1;
5040, 9360, 7920, 3960, 1260, 252, 28, 1;
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MATHEMATICA
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Table[Binomial[2n-k, k](n-k)!, {n, 0, 10}, {k, 0, n}]//Flatten (* Harvey P. Dale, Mar 24 2017 *)
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PROG
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(Sage) flatten([[factorial(n-k)*binomial(2*n-k, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021
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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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