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A268442 Triangle read by rows, the coefficients of the inverse Bell polynomials.

%I A268442 #19 Feb 09 2016 03:44:10
%S A268442 1,0,1,0,-1,1,0,3,-1,-3,1,0,-15,10,-1,15,-4,-6,1,0,105,-105,10,15,-1,
%T A268442 -105,60,-5,45,-10,-10,1,0,-945,1260,-280,-210,35,21,-1,945,-840,70,
%U A268442 105,-6,-420,210,-15,105,-20,-15,1
%N A268442 Triangle read by rows, the coefficients of the inverse Bell polynomials.
%C A268442 The triangle of coefficients of the Bell polynomials is A268441. For the definition of the inverse Bell polynomials see the link 'Bell transform'.
%H A268442 Peter Luschny, <a href="/A268442/b268442.txt">First 21 rows, flattened</a>
%H A268442 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%H A268442 Peter Luschny, <a href="/A268442/a268442.txt">SageMath implementation</a>
%e A268442 [[1]],
%e A268442 [[0], [1]],
%e A268442 [[0], [-1],                    [1]],
%e A268442 [[0], [3, -1],                 [-3],           [1]],
%e A268442 [[0], [-15, 10, -1],           [15, -4],       [-6],      [1]],
%e A268442 [[0], [105, -105, 10, 15, -1], [-105, 60, -5], [45, -10], [-10], [1]]
%e A268442 Replacing the sublists by their sums reduces the triangle to the triangle of the Stirling numbers of first kind (A048994). The column 1 of sublists is A176740 (missing the leading 1) and A134685 in different order.
%t A268442 A268442Matrix[dim_] := Module[ {v, r, A},
%t A268442 v = Table[Subscript[x,j],{j,1,dim}];
%t A268442 r = Table[Subscript[x,j]->1,{j,1,n}];
%t A268442 A = Table[Table[BellY[n,k,v], {k,0,dim}], {n,0,dim}];
%t A268442 Table[Table[MonomialList[Inverse[A][[n,k]]/. r[[1]],
%t A268442 v, Lexicographic] /. r, {k,1,n}] // Flatten, {n,1,dim}]];
%t A268442 A268442Matrix[7] // Flatten
%o A268442 (Sage) # see link
%Y A268442 Cf. A036040, A048994, A134685, A176740, A268441.
%K A268442 sign,tabf
%O A268442 0,8
%A A268442 _Peter Luschny_, Feb 06 2016