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A375597 Triangle read by rows: T(n, k) = n! * 3^k * hypergeom([-k], [-n], -2/3).

%I A375597 #6 Aug 21 2024 05:34:54
%S A375597 1,1,1,2,4,10,6,14,34,82,24,60,152,388,1000,120,312,816,2144,5656,
%T A375597 14968,720,1920,5136,13776,37040,99808,269488,5040,13680,37200,101328,
%U A375597 276432,755216,2066032,5659120,40320,110880,305280,841440,2321664,6412128,17725952,49045792,135819136
%N A375597 Triangle read by rows: T(n, k) = n! * 3^k * hypergeom([-k], [-n], -2/3).
%F A375597 T(n, k) = (-2)^k*Sum_{j=0..k} (-3/2)^(k - j)*binomial(k, k - j)*(n - j)!.
%e A375597 Triangle starts:
%e A375597 [0] 1;
%e A375597 [1] 1, 1;
%e A375597 [2] 2, 4, 10;
%e A375597 [3] 6, 14, 34, 82;
%e A375597 [4] 24, 60, 152, 388, 1000;
%e A375597 [5] 120, 312, 816, 2144, 5656, 14968;
%e A375597 [6] 720, 1920, 5136, 13776, 37040, 99808, 269488;
%e A375597 [7] 5040, 13680, 37200, 101328, 276432, 755216, 2066032, 5659120;
%e A375597 ...
%t A375597 T[n_, k_] := (-2)^k*Sum[(-3/2)^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}];
%t A375597 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
%Y A375597 Cf. A375600, A000142.
%Y A375597 Cf. A374427, A374428, A375446, A375447.
%K A375597 nonn,tabl
%O A375597 0,4
%A A375597 _Detlef Meya_, Aug 20 2024