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A265313 Square array read by ascending antidiagonals, complementary Bell numbers iterated by the Bell transform.

%I A265313 #19 Mar 27 2020 13:57:49
%S A265313 1,1,1,1,-1,1,1,-1,0,1,1,-1,2,1,1,1,-1,2,-4,1,1,1,-1,2,-6,9,-2,1,1,-1,
%T A265313 2,-6,22,-22,-9,1,1,-1,2,-6,24,-95,54,-9,1,1,-1,2,-6,24,-118,472,-139,
%U A265313 50,1,1,-1,2,-6,24,-120,683,-2638,372,267,1,1,-1,2,-6,24
%N A265313 Square array read by ascending antidiagonals, complementary Bell numbers iterated by the Bell transform.
%H A265313 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%e A265313 [ 1,  1, 1,  1,  1,   1,    1,     1,     1, ...] A000012
%e A265313 [ 1, -1, 0,  1,  1,  -2,   -9,    -9,    50, ...] A000587
%e A265313 [ 1, -1, 2, -4,  9, -22,   54,  -139,   372, ...] A265023
%e A265313 [ 1, -1, 2, -6, 22, -95,  472, -2638, 16343, ...]
%e A265313 [ 1, -1, 2, -6, 24, -118, 683, -4533, 33862, ...]
%e A265313 [ 1, -1, 2, -6, 24, -120, 718, -4989, 39405, ...]
%e A265313 [...                                         ...]
%e A265313 [ 1, -1, 2, -6, 24, -120, 720, -5040, 40320, ...] A133942
%o A265313 (Sage) # uses[bell_transform from A264428]
%o A265313 def complementary_bell_number_matrix(ord, len):
%o A265313     b = [1]*len; L = [b]
%o A265313     for k in (1..ord-1):
%o A265313         b = [sum((-1)^n*c for (n, c) in enumerate(bell_transform(n, b))) for n in range(len)]
%o A265313         L.append(b)
%o A265313     return matrix(ZZ, L)
%o A265313 print(complementary_bell_number_matrix(6,9))
%Y A265313 Cf. A000012, A000587, A133942, A264428, A265023, A265312.
%K A265313 sign,tabl
%O A265313 0,13
%A A265313 _Peter Luschny_, Dec 06 2015