A144446 Array t(n, k) = (k*(n-1) +2-k)*t(n-1, k) + k*t(n-2, k), with t(1, k) = 1, t(2, k) = 2, read by antidiagonals.

1, 2, 1, 7, 2, 1, 30, 10, 2, 1, 157, 64, 13, 2, 1, 972, 532, 110, 16, 2, 1, 6961, 5448, 1249, 168, 19, 2, 1, 56660, 66440, 17816, 2416, 238, 22, 2, 1, 516901, 941056, 306619, 44160, 4141, 320, 25, 2, 1, 5225670, 15189776, 6185828, 981184, 92292, 6532, 414, 28, 2, 1

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Oct 05 2008

Keywords

• nonn
• tabl

Examples

			Array t(n,k) begins as:
    1,    1,     1,     1,     1,      1, ...;
    2,    2,     2,     2,     2,      2, ...;
    7,   10,    13,    16,    19,     22, ...;
   30,   64,   110,   168,   238,    320, ...;
  157,  532,  1249,  2416,  4141,   6532, ...;
  972, 5448, 17816, 44160, 92292, 171752, ...;
Antidiagonal triangle T(n,k) begins as:
        1;
        2,        1;
        7,        2,       1;
       30,       10,       2,      1;
      157,       64,      13,      2,     1;
      972,      532,     110,     16,     2,    1;
     6961,     5448,    1249,    168,    19,    2,   1;
    56660,    66440,   17816,   2416,   238,   22,   2,  1;
   516901,   941056,  306619,  44160,  4141,  320,  25,  2, 1;
  5225670, 15189776, 6185828, 981184, 92292, 6532, 414, 28, 2, 1;
		

Links

• G. C. Greubel, Antidiagonals n = 1..50, flattened

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144441, A144442, A144443, A144444, A144445, A144447.

Cf. A001053.

Programs

Magma

function T(n,k) // triangle form; A144446
  if k gt n-2 then return n-k+1;
  else return (k*(n-k)+2-k)*T(n-1, k) + k*T(n-2, k);
  end if; return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 05 2022

Mathematica

t[n_, k_]:= t[n, k]= If[n<3, n, (k*(n-1) +2-k)*t[n-1,k] + k*t[n-2,k]];
T[n_, k_]:= t[n-k+1,k];
Table[T[n, k], {n, 12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 05 2022 *)

Sage

def t(n,k): return n if(n<3) else (k*(n-1) +2-k)*t(n-1, k) + k*t(n-2, k)
def A144446(n,k): return t(n-k+1,k)
flatten([[A144446(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 05 2022

Formula

T(n, k) = t(n-k+1, k), where t(n, k) = (k*(n-1) +2-k)*t(n-1, k) + k*t(n-2, k) with t(1, k) = 1, t(2, k) = 2.

T(n, 1) = A001053(n+1).

T(n, k) = (k*(n-k)+2-k)*T(n-1, k) + k*T(n-2, k) with T(n, n-1) = 2, T(n, n) = 1 (as a triangle). - G. C. Greubel, Mar 05 2022

Extensions

Edited by G. C. Greubel, Mar 05 2022