cp's OEIS Frontend

A265605 Triangle read by rows: The inverse Bell transform of the triple factorial numbers (A007559).

%I A265605 #27 Jun 16 2023 16:22:24
%S A265605 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 A265605 -945,315,-35,0,70,21,1,0,10395,-3465,490,-35,70,154,28,1,0,-135135,
%U A265605 45045,-6895,630,-105,378,294,36,1
%N A265605 Triangle read by rows: The inverse Bell transform of the triple factorial numbers (A007559).
%H A265605 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%H A265605 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 A265605 [ 1]
%e A265605 [ 0,    1]
%e A265605 [ 0,    1,    1]
%e A265605 [ 0,   -1,    3,    1]
%e A265605 [ 0,    3,   -1,    6,    1]
%e A265605 [ 0,  -15,    5,    5,   10,    1]
%e A265605 [ 0,  105,  -35,    0,   25,   15,    1]
%e A265605 [ 0, -945,  315,  -35,    0,   70,   21,    1]
%o A265605 (Sage) # uses[bell_transform from A264428]
%o A265605 def inverse_bell_matrix(generator, dim):
%o A265605     G = [generator(k) for k in srange(dim)]
%o A265605     row = lambda n: bell_transform(n, G)
%o A265605     M = matrix(ZZ, [row(n)+[0]*(dim-n-1) for n in srange(dim)]).inverse()
%o A265605     return matrix(ZZ, dim, lambda n,k: (-1)^(n-k)*M[n,k])
%o A265605 multifact_3_1 = lambda n: prod(3*k + 1 for k in (0..n-1))
%o A265605 print(inverse_bell_matrix(multifact_3_1, 8))
%Y A265605 Cf. A007559, A264428, A264429.
%Y A265605 Inverse Bell transforms of other multifactorials are: A048993, A049404, A049410, A075497, A075499, A075498, A119275, A122848, A265604.
%K A265605 sign,tabl
%O A265605 0,9
%A A265605 _Peter Luschny_, Dec 30 2015