%I #19 Feb 09 2016 03:44:10

%S 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 -105,60,-5,45,-10,-10,1,0,-945,1260,-280,-210,35,21,-1,945,-840,70,

%U 105,-6,-420,210,-15,105,-20,-15,1

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

%C The triangle of coefficients of the Bell polynomials is A268441. For the definition of the inverse Bell polynomials see the link 'Bell transform'.

%H Peter Luschny, <a href="/A268442/b268442.txt">First 21 rows, flattened</a>

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

%H Peter Luschny, <a href="/A268442/a268442.txt">SageMath implementation</a>

%e [[1]],

%e [[0], [1]],

%e [[0], [-1], [1]],

%e [[0], [3, -1], [-3], [1]],

%e [[0], [-15, 10, -1], [15, -4], [-6], [1]],

%e [[0], [105, -105, 10, 15, -1], [-105, 60, -5], [45, -10], [-10], [1]]

%e 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 A268442Matrix[dim_] := Module[ {v, r, A},

%t v = Table[Subscript[x,j],{j,1,dim}];

%t r = Table[Subscript[x,j]->1,{j,1,n}];

%t A = Table[Table[BellY[n,k,v], {k,0,dim}], {n,0,dim}];

%t Table[Table[MonomialList[Inverse[A][[n,k]]/. r[[1]],

%t v, Lexicographic] /. r, {k,1,n}] // Flatten, {n,1,dim}]];

%t A268442Matrix[7] // Flatten

%o (Sage) # see link

%Y Cf. A036040, A048994, A134685, A176740, A268441.

%K sign,tabf

%O 0,8

%A _Peter Luschny_, Feb 06 2016