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A386789 Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).

%I A386789 #13 Aug 06 2025 18:09:33
%S A386789 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,
%T A386789 2100,2310,924,0,8,216,1800,6600,11880,10296,3432,0,9,315,3465,17325,
%U A386789 45045,63063,45045,12870,0,10,440,6160,40040,140140,280280,320320,194480,48620
%N A386789 Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).
%H A386789 Paolo Xausa, <a href="/A386789/b386789.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).
%F A386789 A357367(n, k) = n!*T(n, k).
%e A386789 Triangle begins:
%e A386789   [0] 1;
%e A386789   [1] 0, 2;
%e A386789   [2] 0, 3,   6;
%e A386789   [3] 0, 4,  20,   20;
%e A386789   [4] 0, 5,  45,  105,   70;
%e A386789   [5] 0, 6,  84,  336,  504,   252;
%e A386789   [6] 0, 7, 140,  840, 2100,  2310,   924;
%e A386789   [7] 0, 8, 216, 1800, 6600, 11880, 10296, 3432;
%e A386789 .
%e A386789 Seen as an array A(n, k) = binomial(n + k - 1, n)*binomial(n + 2*k, k):
%e A386789   [0] 1, 2,   6,   20,    70,    252,     924, ... [A000984]
%e A386789   [1] 0, 3,  20,  105,   504,   2310,   10296, ... [A000917]
%e A386789   [2] 0, 4,  45,  336,  2100,  11880,   63063, ...
%e A386789   [3] 0, 5,  84,  840,  6600,  45045,  280280, ...
%e A386789   [4] 0, 6, 140, 1800, 17325, 140140, 1009008, ...
%e A386789   [5] 0, 7, 216, 3465, 40040, 378378, 3118752, ...
%e A386789   [6] 0, 8, 315, 6160, 84084, 917280, 8576568, ...
%p A386789 T := (n, k) -> binomial(n - 1, k - 1)*binomial(n + k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
%t A386789 A386789[n_, k_] := Binomial[n - 1, k - 1]*Binomial[n + k, k];
%t A386789 Table[A386789[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Aug 06 2025 *)
%Y A386789 Cf. A176479 (row sums), A000984 (main diagonal), A181983 (alternating row sums), A386876 (central terms).
%Y A386789 Cf. A000984, A000917, A357367.
%K A386789 nonn,tabl
%O A386789 0,3
%A A386789 _Peter Luschny_, Aug 06 2025