A171830 - OEIS

%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