A375612 - OEIS

%I #6 Aug 30 2024 03:24:00

%S 1,1,3,2,7,25,6,22,81,299,24,90,338,1271,4785,120,456,1734,6598,25121,

%T 95699,720,2760,10584,40602,155810,598119,2296777,5040,19440,75000,

%U 289416,1117062,4312438,16651633,64309755,40320,156240,605520,2347080,9098904,35278554,136801778,530555479,2057912161

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

%F T(n, k) = (-1)^k*Sum_{j=0..k} (-4)^(k - j)*binomial(k, k - j)*(n - j)!.

%e Triangle starts:

%e [0] 1;

%e [1] 1, 3;

%e [2] 2, 7, 25;

%e [3] 6, 22, 81, 299;

%e [4] 24, 90, 338, 1271, 4785;

%e [5] 120, 456, 1734, 6598, 25121, 95699;

%e [6] 720, 2760, 10584, 40602, 155810, 598119, 2296777;

%e [7] 5040, 19440, 75000, 289416, 1117062, 4312438, 16651633, 64309755;

%e ...

%t T[n_, k_] := (-1)^k*Sum[(-4)^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}];

%t Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

%Y Cf. A375613, A000142, A001907 (main diagonal).

%Y Cf. A374427, A374428, A375446, A375447, A375597, A375600.

%K nonn,tabl

%O 0,3

%A _Detlef Meya_, Aug 21 2024