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%I #21 Jul 03 2020 14:55:05
%S 1,0,1,0,1,1,0,-2,3,1,0,10,-5,6,1,0,-80,30,-5,10,1,0,880,-290,45,5,15,
%T 1,0,-12320,3780,-560,35,35,21,1,0,209440,-61460,8820,-735,0,98,28,1,
%U 0,-4188800,1192800,-167300,14700,-735,0,210,36,1
%N Triangle read by rows: The inverse Bell transform of the quartic factorial numbers (A007696).
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%H Richell O. Celeste, Roberto B. Corcino, 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, -2, 3, 1]
%e [ 0, 10, -5, 6, 1]
%e [ 0, -80, 30, -5, 10, 1]
%e [ 0, 880, -290, 45, 5, 15, 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_4_1 = lambda n: prod(4*k + 1 for k in (0..n-1))
%o print(inverse_bell_matrix(multifact_4_1, 8))
%Y Cf. A007696, A264428, A264429.
%Y Inverse Bell transforms of other multifactorials are: A048993, A049404, A049410, A075497, A075499, A075498, A119275, A122848, A265605.
%K sign,tabl
%O 0,8
%A _Peter Luschny_, Dec 30 2015
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