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

0,9

Comments

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

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

Formula

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

Example

[[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).

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 *)

Prog

(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))

Crossrefs

Cf. A048993, A268442.

Sequence in context: A320476 A304326 A099905 * A264435 A356656 A085391

Adjacent sequences: A268438 A268439 A268440 * A268442 A268443 A268444

Keyword

nonn,tabf

Author

Peter Luschny, Feb 07 2016

Status

approved