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A265604 Triangle read by rows: The inverse Bell transform of the quartic factorial numbers (A007696).

%I A265604 #25 Jul 31 2025 07:34:04
%S A265604 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 A265604 1,0,-12320,3780,-560,35,35,21,1,0,209440,-61460,8820,-735,0,98,28,1,
%U A265604 0,-4188800,1192800,-167300,14700,-735,0,210,36,1
%N A265604 Triangle read by rows: The inverse Bell transform of the quartic factorial numbers (A007696).
%H A265604 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%H A265604 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 A265604 [ 1]
%e A265604 [ 0,      1]
%e A265604 [ 0,      1,      1]
%e A265604 [ 0,     -2,      3,      1]
%e A265604 [ 0,     10,     -5,      6,      1]
%e A265604 [ 0,    -80,     30,     -5,     10,      1]
%e A265604 [ 0,    880,   -290,     45,      5,     15,      1]
%o A265604 (Sage) # uses[bell_transform from A264428]
%o A265604 def inverse_bell_matrix(generator, dim):
%o A265604     G = [generator(k) for k in srange(dim)]
%o A265604     row = lambda n: bell_transform(n, G)
%o A265604     M = matrix(ZZ, [row(n)+[0]*(dim-n-1) for n in srange(dim)]).inverse()
%o A265604     return matrix(ZZ, dim, lambda n,k: (-1)^(n-k)*M[n,k])
%o A265604 multifact_4_1 = lambda n: prod(4*k + 1 for k in (0..n-1))
%o A265604 print(inverse_bell_matrix(multifact_4_1, 8))
%Y A265604 Cf. A007696, A264428, A264429.
%Y A265604 Inverse Bell transforms of other multifactorials are: A048993, A049404, A049410, A075497, A075499, A075498, A119275, A122848, A265605.
%K A265604 sign,tabl
%O A265604 0,8
%A A265604 _Peter Luschny_, Dec 30 2015