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Triangle read by rows: The Bell transform of the quartic factorial numbers (A007696).
4

%I #22 Jul 03 2020 14:55:05

%S 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 145,15,1,0,208845,89775,19285,2730,280,21,1,0,5221125,2183895,456190,

%U 62965,6370,490,28,1,0,151412625,62002395,12676265,1715490,171255,13230,798,36,1

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

%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.

%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>

%e [1],

%e [0, 1],

%e [0, 1, 1],

%e [0, 5, 3, 1],

%e [0, 45, 23, 6, 1],

%e [0, 585, 275, 65, 10, 1],

%e [0, 9945, 4435, 990, 145, 15, 1],

%e [0, 208845, 89775, 19285, 2730, 280, 21, 1],

%t (* The function BellMatrix is defined in A264428. *)

%t rows = 10;

%t M = BellMatrix[Pochhammer[1/4, #] 4^# &, rows];

%t Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 23 2019 *)

%o (Sage) # uses[bell_transform from A264428]

%o def A265606_row(n):

%o multifact_4_1 = lambda n: prod(4*k + 1 for k in (0..n-1))

%o mfact = [multifact_4_1(k) for k in (0..n)]

%o return bell_transform(n, mfact)

%o [A265606_row(n) for n in (0..7)]

%Y Cf. A007696, A264428, A264429.

%Y Bell transforms of other multifactorials are: A000369, A004747, A039683, A051141, A051142, A119274, A132062, A132393, A203412.

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Dec 30 2015