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A375600 Triangle read by rows: T(n, k) = n! * 3^k * hypergeom([-k], [-n], 2/3).

%I A375600 #5 Aug 21 2024 05:34:59
%S A375600 1,1,5,2,8,34,6,22,82,314,24,84,296,1052,3784,120,408,1392,4768,16408,
%T A375600 56792,720,2400,8016,26832,90032,302912,1022320,5040,16560,54480,
%U A375600 179472,592080,1956304,6474736,21468848,40320,131040,426240,1387680,4521984,14750112,48162944,157438304,515252608
%N A375600 Triangle read by rows: T(n, k) = n! * 3^k * hypergeom([-k], [-n], 2/3).
%F A375600 T(n, k) = 2^k*Sum_{j=0..k} (3/2)^(k - j)*binomial(k, k - j)*(n - j)!.
%e A375600 Triangle starts:
%e A375600 [0] 1;
%e A375600 [1] 1, 5;
%e A375600 [2] 2, 8, 34;
%e A375600 [3] 6, 22, 82, 314;
%e A375600 [4] 24, 84, 296, 1052, 3784;
%e A375600 [5] 120, 408, 1392, 4768, 16408, 56792;
%e A375600 [6] 720, 2400, 8016, 26832, 90032, 302912, 1022320;
%e A375600 [7] 5040, 16560, 54480, 179472, 592080, 1956304, 6474736, 21468848;
%e A375600 ...
%t A375600 T[n_, k_] := 2^k*Sum[(3/2)^(k - j)*Binomial[k, k - j]*((n - j)!), {j, 0, k}];
%t A375600 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
%Y A375600 Cf. A375597, A000142, A097817 (main diagonal).
%Y A375600 Cf. A374427, A374428, A375446, A375447.
%K A375600 nonn,tabl
%O A375600 0,3
%A A375600 _Detlef Meya_, Aug 20 2024