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A265606 Triangle read by rows: The Bell transform of the quartic factorial numbers (A007696).

%I A265606 #25 Jul 31 2025 07:01:38
%S A265606 1,0,1,0,1,1,0,5,3,1,0,45,23,6,1,0,585,275,65,10,1,0,9945,4435,990,
%T A265606 145,15,1,0,208845,89775,19285,2730,280,21,1,0,5221125,2183895,456190,
%U A265606 62965,6370,490,28,1,0,151412625,62002395,12676265,1715490,171255,13230,798,36,1
%N A265606 Triangle read by rows: The Bell transform of the quartic factorial numbers (A007696).
%H A265606 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.
%H A265606 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%e A265606 [1],
%e A265606 [0, 1],
%e A265606 [0, 1, 1],
%e A265606 [0, 5, 3, 1],
%e A265606 [0, 45, 23, 6, 1],
%e A265606 [0, 585, 275, 65, 10, 1],
%e A265606 [0, 9945, 4435, 990, 145, 15, 1],
%e A265606 [0, 208845, 89775, 19285, 2730, 280, 21, 1],
%t A265606 (* The function BellMatrix is defined in A264428. *)
%t A265606 rows = 10;
%t A265606 M = BellMatrix[Pochhammer[1/4, #] 4^# &, rows];
%t A265606 Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 23 2019 *)
%o A265606 (Sage) # uses[bell_transform from A264428]
%o A265606 def A265606_row(n):
%o A265606     multifact_4_1 = lambda n: prod(4*k + 1 for k in (0..n-1))
%o A265606     mfact = [multifact_4_1(k) for k in (0..n)]
%o A265606     return bell_transform(n, mfact)
%o A265606 [A265606_row(n) for n in (0..7)]
%Y A265606 Cf. A007696, A264428, A264429.
%Y A265606 Bell transforms of other multifactorials are: A000369, A004747, A039683, A051141, A051142, A119274, A132062, A132393, A203412.
%K A265606 nonn,tabl
%O A265606 0,8
%A A265606 _Peter Luschny_, Dec 30 2015