OFFSET
0,2
REFERENCES
Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 165-66
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
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.
EXAMPLE
Triangle begins as:
1;
2, 2;
3, 4, 3;
16, 24, 24, 16;
45, 144, 162, 144, 45;
192, 480, 1152, 1152, 480, 192;
1050, 2400, 4500, 9600, 4500, 2400, 1050;
6912, 15120, 25920, 43200, 43200, 25920, 15120, 6912;
52920, 112896, 185220, 282240, 220500, 282240, 185220, 112896, 52920;
MATHEMATICA
c[n_]:= If[n<=2, 1, (n-3)!]; f[n_]:= (c[n]/(n*c[n-1]));
T[n_, k_]:= c[n+2]*(n+2)*f[n+2]/(f[n-k+1]*f[k+1]);
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 29 2021 *)
PROG
(Sage)
@CachedFunction
def c(n): return 1 if (n<3) else factorial(n-3)
def f(n): return c(n)/(n*c(n-1))
def T(n, k): return (n+2)*c(n+2)*f(n+2)/(f(k+1)*f(n-k+1))
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 29 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Dec 19 2009
EXTENSIONS
Edited by G. C. Greubel, Apr 29 2021
STATUS
approved