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A375612 Triangle read by rows: T(n, k) = n! * 4^k * hypergeom([-k], [-n], -1/4).

%I A375612 #6 Aug 30 2024 03:24:00
%S A375612 1,1,3,2,7,25,6,22,81,299,24,90,338,1271,4785,120,456,1734,6598,25121,
%T A375612 95699,720,2760,10584,40602,155810,598119,2296777,5040,19440,75000,
%U A375612 289416,1117062,4312438,16651633,64309755,40320,156240,605520,2347080,9098904,35278554,136801778,530555479,2057912161
%N A375612 Triangle read by rows: T(n, k) = n! * 4^k * hypergeom([-k], [-n], -1/4).
%F A375612 T(n, k) = (-1)^k*Sum_{j=0..k} (-4)^(k - j)*binomial(k, k - j)*(n - j)!.
%e A375612 Triangle starts:
%e A375612 [0] 1;
%e A375612 [1] 1, 3;
%e A375612 [2] 2, 7, 25;
%e A375612 [3] 6, 22, 81, 299;
%e A375612 [4] 24, 90, 338, 1271, 4785;
%e A375612 [5] 120, 456, 1734, 6598, 25121, 95699;
%e A375612 [6] 720, 2760, 10584, 40602, 155810, 598119, 2296777;
%e A375612 [7] 5040, 19440, 75000, 289416, 1117062, 4312438, 16651633, 64309755;
%e A375612 ...
%t A375612 T[n_, k_] := (-1)^k*Sum[(-4)^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}];
%t A375612 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
%Y A375612 Cf. A375613, A000142, A001907 (main diagonal).
%Y A375612 Cf. A374427, A374428, A375446, A375447, A375597, A375600.
%K A375612 nonn,tabl
%O A375612 0,3
%A A375612 _Detlef Meya_, Aug 21 2024