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A264429 Triangle read by rows, inverse Bell transform of Bell numbers.

%I A264429 #21 Mar 26 2020 15:53:43
%S A264429 1,0,1,0,-1,1,0,1,-3,1,0,0,7,-6,1,0,-5,-10,25,-10,1,0,18,-20,-75,65,
%T A264429 -15,1,0,-7,231,70,-315,140,-21,1,0,-338,-840,1064,945,-980,266,-28,1,
%U A264429 0,2215,-1278,-8918,1512,4935,-2520,462,-36,1
%N A264429 Triangle read by rows, inverse Bell transform of Bell numbers.
%H A264429 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%e A264429 [ 1 ]
%e A264429 [ 0,     1 ]
%e A264429 [ 0,    -1,     1 ]
%e A264429 [ 0,     1,    -3,     1 ]
%e A264429 [ 0,     0,     7,    -6,     1 ]
%e A264429 [ 0,    -5,   -10,    25,   -10,     1 ]
%e A264429 [ 0,    18,   -20,   -75,    65,   -15,     1 ]
%e A264429 [ 0,    -7,   231,    70,  -315,   140,   -21,     1 ]
%e A264429 [ 0,  -338,  -840,  1064,   945,  -980,   266,   -28,     1 ]
%e A264429 [ 0,  2215, -1278, -8918,  1512,  4935, -2520,   462,   -36,   1 ]
%t A264429 rows = 10;
%t A264429 M = Table[BellY[n, k, BellB[Range[0, rows-1]]],{n, 0, rows-1}, {k, 0, rows-1}] // Inverse;
%t A264429 A264429 = Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018 *)
%o A264429 (Sage) # uses[bell_transform from A264428]
%o A264429 def inverse_bell_transform(dim, L):
%o A264429     M = matrix(ZZ, dim)
%o A264429     for n in range(dim):
%o A264429         row = bell_transform(n, L)
%o A264429         for k in (0..n): M[n,k] = row[k]
%o A264429     return M.inverse()
%o A264429 def A264429_matrix(dim):
%o A264429     uno = [1]*dim
%o A264429     bell_numbers = [sum(bell_transform(n, uno)) for n in range(dim)]
%o A264429     return inverse_bell_transform(dim, bell_numbers)
%o A264429 A264429_matrix(10)
%Y A264429 Cf. A000110, A264428.
%K A264429 sign,tabl
%O A264429 0,9
%A A264429 _Peter Luschny_, Nov 13 2015