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%I #27 Jun 16 2023 16:22:24
%S 1,0,1,0,1,1,0,-1,3,1,0,3,-1,6,1,0,-15,5,5,10,1,0,105,-35,0,25,15,1,0,
%T -945,315,-35,0,70,21,1,0,10395,-3465,490,-35,70,154,28,1,0,-135135,
%U 45045,-6895,630,-105,378,294,36,1
%N Triangle read by rows: The inverse Bell transform of the triple factorial numbers (A007559).
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%H Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Celeste/celeste3.html">Two Approaches to Normal Order Coefficients</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
%e [ 1]
%e [ 0, 1]
%e [ 0, 1, 1]
%e [ 0, -1, 3, 1]
%e [ 0, 3, -1, 6, 1]
%e [ 0, -15, 5, 5, 10, 1]
%e [ 0, 105, -35, 0, 25, 15, 1]
%e [ 0, -945, 315, -35, 0, 70, 21, 1]
%o (Sage) # uses[bell_transform from A264428]
%o def inverse_bell_matrix(generator, dim):
%o G = [generator(k) for k in srange(dim)]
%o row = lambda n: bell_transform(n, G)
%o M = matrix(ZZ, [row(n)+[0]*(dim-n-1) for n in srange(dim)]).inverse()
%o return matrix(ZZ, dim, lambda n,k: (-1)^(n-k)*M[n,k])
%o multifact_3_1 = lambda n: prod(3*k + 1 for k in (0..n-1))
%o print(inverse_bell_matrix(multifact_3_1, 8))
%Y Cf. A007559, A264428, A264429.
%Y Inverse Bell transforms of other multifactorials are: A048993, A049404, A049410, A075497, A075499, A075498, A119275, A122848, A265604.
%K sign,tabl
%O 0,9
%A _Peter Luschny_, Dec 30 2015