A326659 T(n,k) = [0=0]; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

1, 1, 1, 1, 4, 2, 1, 15, 18, 6, 1, 64, 132, 96, 24, 1, 325, 980, 1140, 600, 120, 1, 1956, 7830, 12720, 10440, 4320, 720, 1, 13699, 68502, 143850, 162120, 103320, 35280, 5040, 1, 109600, 657608, 1698816, 2447760, 2123520, 1108800, 322560, 40320

Offset: 0

Author

Alois P. Heinz, Sep 12 2019

Keywords

• nonn
• tabl

Comments

[] is an Iverson bracket.

Examples

			Triangle T(n,k) begins:
  1;
  1,     1;
  1,     4,     2;
  1,    15,    18,      6;
  1,    64,   132,     96,     24;
  1,   325,   980,   1140,    600,    120;
  1,  1956,  7830,  12720,  10440,   4320,   720;
  1, 13699, 68502, 143850, 162120, 103320, 35280, 5040;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Wikipedia, Iverson bracket

Crossrefs

Columns k=0-2 give: A000012, A007526, 2*A134432(n-1).

Main diagonal gives A000142.

Row sums give A308876.

Cf. A000522, A073474, A093964, A143409, A196347, A271705, A327606.

Programs

Maple

T:= proc(n, k) option remember;
      `if`(0=0, 1, 0)
    end:
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

T[n_ /; n >= 0, k_ /; k >= 0] := T[n, k] = Boole[0 < k <= n]*n*(T[n-1, k-1] + T[n-1, k]) + Boole[k == 0 && n >= 0];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 09 2021 *)

Formula

E.g.f. of column k: exp(x)*(x/(1-x))^k.

T(n,k) = k! * A271705(n,k).

T(n,k) = n * A073474(n-1,k-1) for n,k >= 1.

T(n,1) = n * A000522(n-1) for n >= 1.

T(n,2) = n * A093964(n-1) for n >= 1.

Sum_{k=1..n} k * T(n,k) = A327606(n).