A326659 T(n,k) = [0<k<=n] * n*(T(n-1,k-1)+T(n-1,k)) + [k=0 and n>=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

0,5

Comments

[] is an Iverson bracket.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Wikipedia, Iverson bracket

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).

Example

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;
  ...

Maple

T:= proc(n, k) option remember;
      `if`(0<k and k<=n, n*(T(n-1, k-1)+T(n-1, k)), 0)+
      `if`(k=0 and n>=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 *)

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.

Sequence in context: A225476 A143777 A365566 * A236830 A269736 A264535

Adjacent sequences: A326656 A326657 A326658 * A326660 A326661 A326662

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 12 2019

Status

approved