A265312 - OEIS

%I #30 Mar 28 2020 10:58:48

%S 1,1,1,1,1,1,1,1,2,1,1,1,2,5,1,1,1,2,6,15,1,1,1,2,6,23,52,1,1,1,2,6,

%T 24,106,203,1,1,1,2,6,24,119,568,877,1,1,1,2,6,24,120,700,3459,4140,1,

%U 1,1,2,6,24,120,719,4748,23544,21147,1,1,1,2,6,24,120,720,5013,36403,176850,115975,1

%N Square array read by ascending antidiagonals, Bell numbers iterated by the Bell transform.

%H Alois P. Heinz, <a href="/A265312/b265312.txt">Antidiagonals n = 0..140, flattened</a>

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

%e [1, 1, 1, 1,  1,   1,   1,    1,     1, ...] A000012

%e [1, 1, 2, 5, 15,  52, 203,  877,  4140, ...] A000110

%e [1, 1, 2, 6, 23, 106, 568, 3459, 23544, ...] A187761

%e [1, 1, 2, 6, 24, 119, 700, 4748, 36403, ...] A264432

%e [1, 1, 2, 6, 24, 120, 719, 5013, 39812, ...]

%e [1, 1, 2, 6, 24, 120, 720, 5039, 40285, ...]

%e [...                                    ...]

%e [1, 1, 2, 6, 24, 120, 720, 5040, 40320, ...] A000142 = main diagonal.

%p A:= proc(n, h) option remember; `if`(min(n, h)=0, 1, add(

%p       binomial(n-1, j-1)*A(j-1, h-1)*A(n-j, h), j=1..n))

%p     end:

%p seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Aug 21 2017

%t A[n_, h_]:=A[n, h]=If[Min[n, h]==0, 1, Sum[Binomial[n - 1, j - 1] A[j - 1, h - 1] A[n - j, h] , {j, n}]]; Table[A[n, d - n], {d, 0, 12}, {n, 0, d}]//Flatten (* _Indranil Ghosh_, Aug 21 2017, after maple code *)

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

%o def bell_number_matrix(ord, len):

%o     b = [1]*len; L = [b]

%o     for k in (1..ord-1):

%o         b = [sum(bell_transform(n, b)) for n in range(len)]

%o         L.append(b)

%o     return matrix(ZZ, L)

%o print(bell_number_matrix(6, 9))

%o (Python)

%o from sympy.core.cache import cacheit

%o from sympy import binomial

%o @cacheit

%o def A(n, h): return 1 if min(n, h)==0 else sum([binomial(n - 1, j - 1)*A(j - 1, h - 1)*A(n - j, h) for j in range(1, n + 1)])

%o for d in range(13): print([A(n, d - n) for n in range(d + 1)]) # _Indranil Ghosh_, Aug 21 2017, after Maple code

%Y Cf. A000012, A000110, A000142, A187761, A264428, A264432, A265313.

%K nonn,tabl

%O 0,9

%A _Peter Luschny_, Dec 06 2015