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A268441
Triangle read by rows, the coefficients of the Bell polynomials.
7
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
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
Sequence in context: A320476 A304326 A099905 * A264435 A356656 A085391
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Feb 07 2016
STATUS
approved