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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.
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%I #20 Mar 31 2022 22:00:07

%S 1,0,1,0,1,2,0,2,9,6,0,6,50,72,24,0,24,350,850,600,120,0,120,3014,

%T 11250,12900,5400,720,0,720,31164,170618,286650,191100,52920,5040,0,

%U 5040,378888,2962736,6909784,6585600,2869440,564480,40320

%N 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.

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

%H Andreas B. G. Blobel, <a href="https://arxiv.org/abs/2203.09519">On convolution powers of 1/x</a>, arXiv:2203.09519 [math.CO], 2022.

%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>

%e [n\k 0 1 2 3 4 5 6 7]

%e [0] [1]

%e [1] [0, 1]

%e [2] [0, 1, 2]

%e [3] [0, 2, 9, 6]

%e [4] [0, 6, 50, 72, 24]

%e [5] [0, 24, 350, 850, 600, 120]

%e [6] [0, 120, 3014, 11250, 12900, 5400, 720]

%e [7] [0, 720, 31164, 170618, 286650, 191100, 52920, 5040]

%t (* The function BellMatrix is defined in A264428 *)

%t nmax = 8;

%t M = BellMatrix[1/(# + 1)&, nmax + 1];

%t B[n_, k_] := M[[n + 1, k + 1]];

%t T[n_, k_] := n! B[n, k];

%t Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 12 2019 *)

%o (Sage) # uses[bell_transform from A264428]

%o def A265607_row(n):

%o invnat = [1/k for k in (1..n)]

%o return [factorial(n)*b for b in bell_transform(n, invnat)]

%o [A265607_row(n) for n in range(9)]

%Y Cf. A264428.

%K nonn,tabl

%O 0,6

%A _Peter Luschny_, Dec 20 2015