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A375613 Triangle read by rows: T(n, k) = n! * 4^k * hypergeom([-k], [-n], 1/4).

%I A375613 #6 Aug 30 2024 03:24:03
%S A375613 1,1,5,2,9,41,6,26,113,493,24,102,434,1849,7889,120,504,2118,8906,
%T A375613 37473,157781,720,3000,12504,52134,217442,907241,3786745,5040,20880,
%U A375613 86520,358584,1486470,6163322,25560529,106028861,40320,166320,686160,2831160,11683224,48219366,199040786,821723673,3392923553
%N A375613 Triangle read by rows: T(n, k) = n! * 4^k * hypergeom([-k], [-n], 1/4).
%F A375613 T(n, k) = Sum_{j=0..k} 4^(k - j)*binomial(k, k - j)*(n - j)!.
%e A375613 Triangle starts:
%e A375613 [0] 1;
%e A375613 [1] 1, 5;
%e A375613 [2] 2, 9, 41;
%e A375613 [3] 6, 26, 113, 493;
%e A375613 [4] 24, 102, 434, 1849, 7889;
%e A375613 [5] 120, 504, 2118, 8906, 37473, 157781;
%e A375613 [6] 720, 3000, 12504, 52134, 217442, 907241, 3786745;
%e A375613 [7] 5040, 20880, 86520, 358584, 1486470, 6163322, 25560529, 106028861;
%e A375613 ...
%t A375613 T[n_, k_] := Sum[4^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}];
%t A375613 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
%Y A375613 Cf. A375612, A000142, A056545 (main diagonal).
%Y A375613 Cf. A374427, A374428, A375446, A375447, A375597, A375600.
%K A375613 nonn,tabl
%O A375613 0,3
%A A375613 _Detlef Meya_, Aug 21 2024