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%I #11 Apr 30 2021 02:15:27
%S 1,2,2,3,4,3,16,24,24,16,45,144,162,144,45,192,480,1152,1152,480,192,
%T 1050,2400,4500,9600,4500,2400,1050,6912,15120,25920,43200,43200,
%U 25920,15120,6912,52920,112896,185220,282240,220500,282240,185220,112896,52920
%N Triangle T(n, k) = (n+2)*c(n+2)*f(n+2)/(f(n-k+1)*f(k+1)) where f(n) = c(n)/(n*c(n-1)), c(n) = (n-3)! for n>2 and 1 otherwise, read by rows.
%D Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 165-66
%H G. C. Greubel, <a href="/A171830/b171830.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = (n+2)*c(n+2)*f(n+2)/(f(n-k+1)*f(k+1)) where f(n) = c(n)/(n*c(n-1)), c(n) = (n-3)! for n>2 and 1 otherwise.
%e Triangle begins as:
%e 1;
%e 2, 2;
%e 3, 4, 3;
%e 16, 24, 24, 16;
%e 45, 144, 162, 144, 45;
%e 192, 480, 1152, 1152, 480, 192;
%e 1050, 2400, 4500, 9600, 4500, 2400, 1050;
%e 6912, 15120, 25920, 43200, 43200, 25920, 15120, 6912;
%e 52920, 112896, 185220, 282240, 220500, 282240, 185220, 112896, 52920;
%t c[n_]:= If[n<=2, 1, (n-3)!]; f[n_]:= (c[n]/(n*c[n-1]));
%t T[n_, k_]:= c[n+2]*(n+2)*f[n+2]/(f[n-k+1]*f[k+1]);
%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 29 2021 *)
%o (Sage)
%o @CachedFunction
%o def c(n): return 1 if (n<3) else factorial(n-3)
%o def f(n): return c(n)/(n*c(n-1))
%o def T(n, k): return (n+2)*c(n+2)*f(n+2)/(f(k+1)*f(n-k+1))
%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 29 2021
%K nonn,tabl,easy,less
%O 0,2
%A _Roger L. Bagula_, Dec 19 2009
%E Edited by _G. C. Greubel_, Apr 29 2021
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