A265607 Triangle read by rows, T(n,k) = n!*B(n,k) for n>=0 and 0<=k<=n, where B(n,k) is the Bell matrix with generator 1/j for j>=1.

1, 0, 1, 0, 1, 2, 0, 2, 9, 6, 0, 6, 50, 72, 24, 0, 24, 350, 850, 600, 120, 0, 120, 3014, 11250, 12900, 5400, 720, 0, 720, 31164, 170618, 286650, 191100, 52920, 5040, 0, 5040, 378888, 2962736, 6909784, 6585600, 2869440, 564480, 40320

Offset

0,6

Comments

See A264428 and the link for the definition of the Bell transform and the Bell matrix.

Links

Table of n, a(n) for n=0..44.

Andreas B. G. Blobel, On convolution powers of 1/x, arXiv:2203.09519 [math.CO], 2022.

Peter Luschny, The Bell transform

Example

[n\k 0    1      2       3       4       5     6     7]
[0] [1]
[1] [0,   1]
[2] [0,   1,     2]
[3] [0,   2,     9,      6]
[4] [0,   6,    50,     72,     24]
[5] [0,  24,   350,    850,    600,    120]
[6] [0, 120,  3014,  11250,  12900,   5400,   720]
[7] [0, 720, 31164, 170618, 286650, 191100, 52920, 5040]

Mathematica

(*  The function BellMatrix is defined in A264428 *)
nmax = 8;
M = BellMatrix[1/(# + 1)&, nmax + 1];
B[n_, k_] := M[[n + 1, k + 1]];
T[n_, k_] := n! B[n, k];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)

Prog

(Sage) # uses[bell_transform from A264428]
def A265607_row(n):
    invnat = [1/k for k in (1..n)]
    return [factorial(n)*b for b in bell_transform(n, invnat)]
[A265607_row(n) for n in range(9)]

Crossrefs

Cf. A264428.

Sequence in context: A259356 A137302 A324305 * A332628 A091518 A096734

Adjacent sequences: A265604 A265605 A265606 * A265608 A265609 A265610

Keyword

nonn,tabl

Author

Peter Luschny, Dec 20 2015

Status

approved