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A265312 Square array read by ascending antidiagonals, Bell numbers iterated by the Bell transform.

%I A265312 #30 Mar 28 2020 10:58:48
%S A265312 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 A265312 24,106,203,1,1,1,2,6,24,119,568,877,1,1,1,2,6,24,120,700,3459,4140,1,
%U A265312 1,1,2,6,24,120,719,4748,23544,21147,1,1,1,2,6,24,120,720,5013,36403,176850,115975,1
%N A265312 Square array read by ascending antidiagonals, Bell numbers iterated by the Bell transform.
%H A265312 Alois P. Heinz, <a href="/A265312/b265312.txt">Antidiagonals n = 0..140, flattened</a>
%H A265312 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%e A265312 [1, 1, 1, 1,  1,   1,   1,    1,     1, ...] A000012
%e A265312 [1, 1, 2, 5, 15,  52, 203,  877,  4140, ...] A000110
%e A265312 [1, 1, 2, 6, 23, 106, 568, 3459, 23544, ...] A187761
%e A265312 [1, 1, 2, 6, 24, 119, 700, 4748, 36403, ...] A264432
%e A265312 [1, 1, 2, 6, 24, 120, 719, 5013, 39812, ...]
%e A265312 [1, 1, 2, 6, 24, 120, 720, 5039, 40285, ...]
%e A265312 [...                                    ...]
%e A265312 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, ...] A000142 = main diagonal.
%p A265312 A:= proc(n, h) option remember; `if`(min(n, h)=0, 1, add(
%p A265312       binomial(n-1, j-1)*A(j-1, h-1)*A(n-j, h), j=1..n))
%p A265312     end:
%p A265312 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Aug 21 2017
%t A265312 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 A265312 (Sage) # uses[bell_transform from A264428]
%o A265312 def bell_number_matrix(ord, len):
%o A265312     b = [1]*len; L = [b]
%o A265312     for k in (1..ord-1):
%o A265312         b = [sum(bell_transform(n, b)) for n in range(len)]
%o A265312         L.append(b)
%o A265312     return matrix(ZZ, L)
%o A265312 print(bell_number_matrix(6, 9))
%o A265312 (Python)
%o A265312 from sympy.core.cache import cacheit
%o A265312 from sympy import binomial
%o A265312 @cacheit
%o A265312 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 A265312 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 A265312 Cf. A000012, A000110, A000142, A187761, A264428, A264432, A265313.
%K A265312 nonn,tabl
%O A265312 0,9
%A A265312 _Peter Luschny_, Dec 06 2015