A171830 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.

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

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

Sequence in context: A164975 A253889 A228754 * A071506 A125920 A176360

Adjacent sequences: A171827 A171828 A171829 * A171831 A171832 A171833

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Dec 19 2009

Extensions

Edited by G. C. Greubel, Apr 29 2021

Status

approved