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A375446 Triangle read by rows: T(n, k) = n! * 3^k * hypergeom([-k], [-n], -1/3).

%I A375446 #16 Aug 19 2024 15:31:18
%S A375446 1,1,2,2,5,13,6,16,43,116,24,66,182,503,1393,120,336,942,2644,7429,
%T A375446 20894,720,2040,5784,16410,46586,132329,376093,5040,14400,41160,
%U A375446 117696,336678,963448,2758015,7897952,40320,115920,333360,958920,2759064,7940514,22858094,65816267,189550849
%N A375446 Triangle read by rows: T(n, k) = n! * 3^k * hypergeom([-k], [-n], -1/3).
%F A375446 T(n, k) = (-1)^k*Sum_{j=0..k} (-3)^(k - j)*binomial(k, k - j)*(n - j)!.
%e A375446 Triangle starts:
%e A375446 [0] 1,
%e A375446 [1] 1, 2,
%e A375446 [2] 2, 5, 13,
%e A375446 [3] 6, 16, 43, 116,
%e A375446 [4] 24, 66, 182, 503, 1393,
%e A375446 [5] 120, 336, 942, 2644, 7429, 20894,
%e A375446 [6] 720, 2040, 5784, 16410, 46586, 132329, 376093,
%e A375446 [7] 5040, 14400, 41160, 117696, 336678, 963448, 2758015, 7897952,
%e A375446 ...
%t A375446 T[n_, k_] := (-1)^k*Sum[(-3)^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}];
%t A375446 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
%Y A375446 Cf. A375447, A000142, A000180 (diagonal).
%Y A375446 Cf. A374427, A374428.
%K A375446 nonn,tabl
%O A375446 0,3
%A A375446 _Detlef Meya_, Aug 15 2024