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A375447 Triangle read by rows: T(n, k) = n! * 3^k * hypergeom([-k], [-n], 1/3).

%I A375447 #14 Aug 19 2024 15:31:11
%S A375447 1,1,4,2,7,25,6,20,67,226,24,78,254,829,2713,120,384,1230,3944,12661,
%T A375447 40696,720,2280,7224,22902,72650,230611,732529,5040,15840,49800,
%U A375447 156624,492774,1550972,4883527,15383110,40320,126000,393840,1231320,3850584,12044526,37684550,117937177,369194641
%N A375447 Triangle read by rows: T(n, k) = n! * 3^k * hypergeom([-k], [-n], 1/3).
%F A375447 T(n, k) = Sum_{j=0..k} 3^(k - j)*binomial(k, k - j)*(n - j)!.
%e A375447 Triangle starts:
%e A375447 [0] 1;
%e A375447 [1] 1, 4;
%e A375447 [2] 2, 7, 25;
%e A375447 [3] 6, 20, 67, 226;
%e A375447 [4] 24, 78, 254, 829, 2713;
%e A375447 [5] 120, 384, 1230, 3944, 12661, 40696;
%e A375447 [6] 720, 2280, 7224, 22902, 72650, 230611, 732529;
%t A375447 T[n_, k_] := Sum[3^(k - j)*Binomial[k, k - j]*((n - j)!), {j, 0, k}];
%t A375447 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
%Y A375447 Cf. A375446, A000142, A010845.
%Y A375447 Cf. A374427, A374428.
%K A375447 nonn,tabl
%O A375447 0,3
%A A375447 _Detlef Meya_, Aug 15 2024