A268441 Triangle read by rows, the coefficients of the Bell polynomials.

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 3, 4, 6, 1, 0, 1, 10, 5, 15, 10, 10, 1, 0, 1, 10, 15, 6, 15, 60, 15, 45, 20, 15, 1, 0, 1, 35, 21, 7, 105, 70, 105, 21, 105, 210, 35, 105, 35, 21, 1, 0, 1, 35, 56, 28, 8, 280, 210, 280, 168, 28, 105, 840, 280, 420, 56, 420, 560, 70, 210, 56, 28, 1

Offset: 0

Author

Peter Luschny, Feb 07 2016

Keywords

• nonn
• tabf

Comments

The triangle of coefficients of the inverse Bell polynomials is A268442.

Examples

			[[1]]
[[0], [1]]
[[0], [1],  [1]]
[[0], [1],  [3],         [1]]
[[0], [1],  [3, 4],      [6],          [1]]
[[0], [1],  [10, 5],     [15, 10],     [10],     [1]]
[[0], [1],  [10, 15, 6], [15, 60, 15], [45, 20], [15], [1]]
Replacing the sublists by their sums reduces the triangle to the triangle of the Stirling numbers of second kind (A048993).
		

References

• L. Comtet, Advanced combinatorics, The art of finite and infinite expansions, 1974.

Links

• Peter Luschny, First 26 rows, flattened
• E. T. Bell, Partition polynomials, Ann. Math., 29 (1927-1928), 38-46.
• E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258-277.
• Peter Luschny, The Bell transform

Crossrefs

Cf. A048993, A268442.

Programs

Mathematica

BellCoeffs[n_, k_] :=  Module[{v, r},
v = Table[Subscript[x,j], {j,1,n}]; (* list of variables *)
r = Table[Subscript[x,j]->1, {j,1,n}]; (* evaluated at 1 *)
MonomialList[BellY[n,k,v], v, NegativeLexicographic] /. r];
A268441Row[n_] := Table[BellCoeffs[n,k], {k,0,n}] // Flatten;
Do[Print[A268441Row[n]], {n,0,8}] (* Peter Luschny, Feb 08 2016 *)
max = 9; egf = Exp[Sum[x[k]*t^k/k!, {k, 1, max}]]; P = Table[n!* SeriesCoefficient[egf, {t, 0, n}], {n, 0, max-1}]; row[n_] := (s = Split[ Sort[{ Exponent[# /. x[] -> x, x], #}& /@ (List @@ Expand[P[[n]]])], #1[[1]] == #2[[1]]&]; Join[{0}, #[[All, 2]]& /@ (s /. x[] -> 1) // Flatten]); row[1] = {1}; Array[row, max] // Flatten (* Jean-François Alcover, Feb 08 2016 *)

Sage

import itertools
def A268441_row(n):
    c = [bell_polynomial(n,k).coefficients() for k in (0..n)]
    if n>0: c[0] = [0]
    return list(itertools.chain(*c))
for n in range(9): print(A268441_row(n))

Formula

E.g.f.: exp( Sum_{k>=1} x_{k}*t^k/k! ), monomials in negative lexicographic order.