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

1,2

Links

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

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

Example

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;

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 *)

Prog

(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
(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

Crossrefs

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

Cf. A001053.

Sequence in context: A128747 A205945 A124392 * A121416 A317547 A089329

Adjacent sequences: A144443 A144444 A144445 * A144447 A144448 A144449

Keyword

nonn,tabl

Author

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

Extensions

Edited by G. C. Greubel, Mar 05 2022

Status

approved