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A118394 Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.

%I A118394 #18 Feb 25 2022 11:00:22
%S A118394 1,1,1,1,6,1,24,1,60,1,120,360,1,210,2520,1,336,10080,1,504,30240,
%T A118394 60480,1,720,75600,604800,1,990,166320,3326400,1,1320,332640,13305600,
%U A118394 19958400,1,1716,617760,43243200,259459200,1,2184,1081080,121080960,1816214400
%N A118394 Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.
%C A118394 Row sums form A118395.
%C A118394 Eigenvector is A118396.
%H A118394 G. C. Greubel, <a href="/A118394/b118394.txt">Rows n = 0..150 of the triangle, flattened</a>
%F A118394 E.g.f.: A(x,y) = exp(x + y*x^3).
%e A118394 Triangle begins:
%e A118394   1;
%e A118394   1;
%e A118394   1;
%e A118394   1,    6;
%e A118394   1,   24;
%e A118394   1,   60;
%e A118394   1,  120,    360;
%e A118394   1,  210,   2520;
%e A118394   1,  336,  10080;
%e A118394   1,  504,  30240,    60480;
%e A118394   1,  720,  75600,   604800;
%e A118394   1,  990, 166320,  3326400;
%e A118394   1, 1320, 332640, 13305600, 19958400;
%e A118394   ...
%t A118394 T[n_, k_] := n!/(k!(n-3k)!);
%t A118394 Table[T[n, k], {n, 0, 14}, {k, 0, Floor[n/3]}] // Flatten (* _Jean-François Alcover_, Nov 04 2020 *)
%o A118394 (PARI) T(n,k)=if(n<3*k || k<0,0,n!/k!/(n-3*k)!)
%o A118394 (Sage)
%o A118394 f=factorial;
%o A118394 flatten([[f(n)/(f(k)*f(n-3*k)) for k in [0..n/3]] for n in [0..20]]) # _G. C. Greubel_, Mar 07 2021
%o A118394 (Magma)
%o A118394 F:= Factorial;
%o A118394 [F(n)/(F(k)*F(n-3*k)): k in [0..Floor(n/3)], n in [0..20]]; // _G. C. Greubel_, Mar 07 2021
%Y A118394 Cf. A118395 (row sums), A118396 (eigenvector).
%Y A118394 Variants: A059344, A118931.
%K A118394 nonn,tabf
%O A118394 0,5
%A A118394 _Paul D. Hanna_, May 07 2006