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

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, -105, 60, -5, 45, -10, -10, 1, 0, -945, 1260, -280, -210, 35, 21, -1, 945, -840, 70, 105, -6, -420, 210, -15, 105, -20, -15, 1

Offset: 0

Author

Peter Luschny, Feb 06 2016

Keywords

• sign
• tabf

Comments

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

Examples

			[[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], [-105, 60, -5], [45, -10], [-10], [1]]
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.
		

Links

• Peter Luschny, First 21 rows, flattened
• Peter Luschny, The Bell transform
• Peter Luschny, SageMath implementation

Crossrefs

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

Programs

Mathematica

A268442Matrix[dim_] := Module[ {v, r, A},
v = Table[Subscript[x,j],{j,1,dim}];
r = Table[Subscript[x,j]->1,{j,1,n}];
A = Table[Table[BellY[n,k,v], {k,0,dim}], {n,0,dim}];
Table[Table[MonomialList[Inverse[A][[n,k]]/. r[[1]],
v, Lexicographic] /. r, {k,1,n}] // Flatten, {n,1,dim}]];
A268442Matrix[7] // Flatten

Sage

# see link