A386789 Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).
1, 0, 2, 0, 3, 6, 0, 4, 20, 20, 0, 5, 45, 105, 70, 0, 6, 84, 336, 504, 252, 0, 7, 140, 840, 2100, 2310, 924, 0, 8, 216, 1800, 6600, 11880, 10296, 3432, 0, 9, 315, 3465, 17325, 45045, 63063, 45045, 12870, 0, 10, 440, 6160, 40040, 140140, 280280, 320320, 194480, 48620
Offset: 0
Examples
Triangle begins: [0] 1; [1] 0, 2; [2] 0, 3, 6; [3] 0, 4, 20, 20; [4] 0, 5, 45, 105, 70; [5] 0, 6, 84, 336, 504, 252; [6] 0, 7, 140, 840, 2100, 2310, 924; [7] 0, 8, 216, 1800, 6600, 11880, 10296, 3432; . Seen as an array A(n, k) = binomial(n + k - 1, n)*binomial(n + 2*k, k): [0] 1, 2, 6, 20, 70, 252, 924, ... [A000984] [1] 0, 3, 20, 105, 504, 2310, 10296, ... [A000917] [2] 0, 4, 45, 336, 2100, 11880, 63063, ... [3] 0, 5, 84, 840, 6600, 45045, 280280, ... [4] 0, 6, 140, 1800, 17325, 140140, 1009008, ... [5] 0, 7, 216, 3465, 40040, 378378, 3118752, ... [6] 0, 8, 315, 6160, 84084, 917280, 8576568, ...
Links
- Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
Crossrefs
Programs
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Maple
T := (n, k) -> binomial(n - 1, k - 1)*binomial(n + k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
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Mathematica
A386789[n_, k_] := Binomial[n - 1, k - 1]*Binomial[n + k, k]; Table[A386789[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Aug 06 2025 *)
Formula
A357367(n, k) = n!*T(n, k).