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A268442 revision #16

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

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OFFSET

0,8

COMMENTS

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

LINKS

Peter Luschny, First 21 rows, flattened

Peter Luschny, The Bell transform

Peter Luschny, SageMath implementation

EXAMPLE

[[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.

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

PROG

(Sage) # see link

CROSSREFS

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