A257563 Triangle read by rows, coefficients T(n,k) of polynomials related to the Bell polynomials, for n>=0 and 0<=k<=n.

1, 0, 2, 0, 1, 3, 0, 1, 5, 4, 0, 1, 10, 14, 5, 0, 1, 19, 48, 30, 6, 0, 1, 36, 149, 158, 55, 7, 0, 1, 69, 445, 727, 413, 91, 8, 0, 1, 134, 1308, 3126, 2638, 924, 140, 9, 0, 1, 263, 3822, 12932, 15396, 7818, 1848, 204, 10, 0, 1, 520, 11159, 52278, 84920, 59382, 19998, 3396, 285, 11

Offset

0,3

Comments

The multivariate partial Bell polynomials as defined for example by L. Comtet, Advanced Combinatorics, depend on the variables x_1, x_2, ... Here we add a variable x_0 to cover the case of the first column (k=0) in the lower triangular matrix of the coefficients of the polynomials in a systematic way. The Bell polynomial is defined in the usual way as the row sum of the partial Bell polynomials and the univariate Bell polynomial as the Bell polynomial with all x_n set to the same variable x. We call these polynomials the x0-based univariate Bell polynomials. The classical univariate Bell polynomials are obtained by substituting x_0 = 0 prior to collapsing the x_n to one variable. In this case the coefficients of the polynomials are the Stirling subset numbers. The triangle given here are the coefficients of the polynomials in the general case.

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 135.

Links

Table of n, a(n) for n=0..65.

Peter Luschny, The Bell Transform

Formula

Row sums are Bell(n)*(2-0^n) = A186021(n).

Example

The triangle starts:
[1]
[0, 2]
[0, 1, 3]
[0, 1, 5, 4]
[0, 1, 10, 14, 5]
[0, 1, 19, 48, 30, 6]
[0, 1, 36, 149, 158, 55, 7]

Maple

T_row := proc(n) local T; T := proc(n, k) option remember; if k = 0 then x^n else add(binomial(n-1, j-1)*T(n-j, k-1)*x, j=0..n-k+1) fi end; PolynomialTools:-CoefficientList(add(T(n, k), k=0..n), x) end: seq(print(T_row(n)), n=0..6);

Mathematica

T[n_, k_] := T[n, k] = If[k==0, x^n, Sum[Binomial[n-1, j-1]*T[n-j, k-1]*x, {j, 0, n-k+1}]]; row[n_] := CoefficientList[Sum[T[n, k], {k, 0, n}], x]; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 20 2016, adapted from Maple *)

Prog

(Sage)
def partial_bell_polynomial(n, k):
    X = var(['x_'+str(i) for i in (0..n+1)])
    @cached_function
    def T(n, k):
        if k==0: return X[0]^n
        return sum(binomial(n-1, j-1)*X[j]*T(n-j, k-1) for j in (0..n-k+1))
    return T(n, k).expand()
def univariate_bell_polynomial(n): # for comparison only
    p = sum(partial_bell_polynomial(n, k) for k in (0..n)).subs(x_0=0)
    q = p({p.variables()[i]:x for i in range(len(p.variables()))})
    R = PolynomialRing(QQ, 'x')
    return R(q)
def x0_based_univariate_bell_polynomial(n):
    p = sum(partial_bell_polynomial(n, k) for k in (0..n))
    q = p({p.variables()[i]:x for i in range(len(p.variables()))})
    R = PolynomialRing(QQ, 'x')
    return R(q)
for n in (0..6): x0_based_univariate_bell_polynomial(n).list()

Crossrefs

Cf. A000110, A048993, A186021.

Sequence in context: A368591 A029297 A289871 * A373118 A219311 A363393

Adjacent sequences: A257560 A257561 A257562 * A257564 A257565 A257566

Keyword

nonn,tabl,easy

Author

Peter Luschny, May 10 2015

Status

approved