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A114494
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Triangle read by rows: T(n,k) is number of hill-free Dyck paths of semilength n and having k returns to the x-axis. (A Dyck path is said to be hill-free if it has no peaks at level 1.)
(history;
published version)
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#26 by Charles R Greathouse IV at Thu Sep 08 08:45:23 EDT 2022
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| PROG
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(MAGMAMagma) /* except 0 as triangle */ [[(k/(n-k))*Binomial(2*n-2*k, n-2*k): k in [1..n div 2]]: n in [2.. 15]]; // Vincenzo Librandi, Sep 15 2018 *)
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Discussion
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Thu Sep 08
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| OEIS Server: https://oeis.org/edit/global/2944
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#25 by Peter Luschny at Sun Sep 16 16:55:39 EDT 2018
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#24 by Joerg Arndt at Sun Sep 16 05:01:57 EDT 2018
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#23 by Jon E. Schoenfield at Sat Sep 15 13:59:46 EDT 2018
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#22 by Jon E. Schoenfield at Sat Sep 15 13:59:43 EDT 2018
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| NAME
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Triangle read by rows: T(n,k) is number of hill-free Dyck paths of semilength n and having k returns to the x-axis. (A Dyck path is said to be hill-free if it has no peaks at level 1)..)
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| STATUS
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reviewed
editing
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#21 by Michel Marcus at Sat Sep 15 09:04:53 EDT 2018
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#20 by Vincenzo Librandi at Sat Sep 15 02:25:05 EDT 2018
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#19 by Vincenzo Librandi at Sat Sep 15 02:24:51 EDT 2018
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| FORMULA
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T(n, k) = (k/(n-k))*binomial(2n2*n-2k2*k, n-2k2*k) (1 <= k <= floor(n/2)).
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| MATHEMATICA
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Join[{0}, t[n_, k_]:=(k/(n - k)) Binomial [2 n - 2 k, n - 2 k]; Table[t[n, k], {n, 1, 20}, {k, n/2}]//Flatten] (* Vincenzo Librandi, Sep 15 2018 *)
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| PROG
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(MAGMA) /* except 0 as triangle */ [[(k/(n-k))*Binomial(2*n-2*k, n-2*k): k in [1..n div 2]]: n in [2.. 15]]; // Vincenzo Librandi, Sep 15 2018 *)
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| STATUS
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reviewed
editing
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#18 by Joerg Arndt at Sat Sep 15 02:00:08 EDT 2018
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#17 by Michel Marcus at Sat Sep 15 01:48:37 EDT 2018
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