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

0,8

Comments

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

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.

Sequence in context: A263526 A132301 A073272 * A175623 A121273 A063065

Adjacent sequences: A268439 A268440 A268441 * A268443 A268444 A268445

Keyword

sign,tabf

Author

Peter Luschny, Feb 06 2016

Status

approved