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A263633 Irregular triangle read by rows: row n gives coefficients of n-th ordinary Bell polynomial B_n(x_1, x_2, ...) with monomials sorted into graded lexicographic order. 7
1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 4, 3, 3, 2, 2, 1, 1, 5, 4, 6, 3, 6, 1, 2, 2, 1, 1, 1, 6, 5, 10, 4, 12, 4, 3, 6, 3, 3, 2, 2, 2, 1, 1, 7, 6, 15, 5, 20, 10, 4, 12, 6, 12, 1, 3, 6, 6, 3, 3, 2, 2, 2, 1, 1, 1, 8, 7, 21, 6, 30, 20, 5, 20, 10, 30, 5, 4, 12, 12, 12, 12, 4, 3, 6, 6, 3, 3, 6, 1, 2, 2, 2, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

"Ordinary" here means in contrast to "exponential", cf. A178867 (see Comtet).

Graded lexicographic order with x[1] > x[2] > ... > x[n] means that monomials are compared first by their total degree, with ties broken by lexicographic order. These monomials correspond to integer partitions.

Row sums are powers of 2. Numbers of terms in rows are partition numbers A000041.

OP_n(-a_1,..,-a_n) = EP_n(a_1,2!*a_2,..,n!*a_n) / n!, where OP_n(a_1,..,a_n) are the partition polynomials of this entry and EP_n, the polynomials of A133314; i.e., the sequences are related as reciprocal o.g.f.s are to reciprocal e.g.f.s. The polynomials play a role in expansion of the iterated Lie derivative (g(x) D_x)^n) formalism for the compositional inversion sketched in A133932. With x[n] = t, the array reduces to the Pascal matrix A007318. - Tom Copeland, Sep 19 2016

The signed row partition polynomials can be generated by the Gram determinants of equation 2.23 on page 133 of the Verde-Star paper. E.g., h_3 = -b_1^3 + 2 b_1 b_2 - b_3 corresponds to the third row. The connection to A133314 is obtained by substituting a(k) = k!*b_k = -k!*x[k] and b(k) = k!*h_k in A133314 to compute reciprocals of o.g.f.s rather than e.g.f.s. - Tom Copeland, Dec 04 2016

For a relation to lambda operations in K-theory on vector bundles, see p. 218 of Dugger. - Tom Copeland, Jul 25 2017

Since E(x) = (1+x_1*x)(1+x_2*x)...(1+x_m*x) is the o.g.f. for the elementary symmetric polynomials e_n(x_1,x_2,...,x_m) and the o.g.f. for the complete symmetric polynomials h_n(x_1,x_2,...,x_m) is H(x) = 1 / E(-x), this entry's partition polynomials with correct signs give either sequence in terms of the other. - Tom Copeland, Jan 29 2018

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 136, 309.

LINKS

Table of n, a(n) for n=1..96.

T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.

T. Copeland, In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms, 2019.

D. Dugger, A Geometric Introduction to K-Theory.

Luis Verde-Star, Representation of symmetric functions as Gram determinants, Advances in Mathematics, 1 Dec 1998, Vol. 140(1):128-143.

Jin Wang, Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.

FORMULA

G.f.: 1/(1-Sum_{i >= 1} x_i*t^i) = 1 + Sum_{n >= 1} B_n(x_1, x_2,...)*t^n. [Comtet, p. 136, Eq. [3o'].]

EXAMPLE

The first few polynomials are:

1, x[1]

2, x[1]^2 + x[2]

3, x[1]^3 + 2*x[1]*x[2] + x[3]

4, x[1]^4 + 3*x[1]^2*x[2] + 2*x[1]*x[3] + x[2]^2 + x[4]

5, x[1]^5 + 4*x[1]^3*x[2] + 3*x[1]^2*x[3] + 3*x[1]*x[2]^2 + 2*x[1]*x[4] + 2*x[2]*x[3] + x[5]

6, x[1]^6 + 5*x[1]^4*x[2] + 4*x[1]^3*x[3] + 6*x[1]^2*x[2]^2 + 3*x[1]^2*x[4] + 6*x[1]*x[2]*x[3] + x[2]^3 + 2*x[1]*x[5] + 2*x[2]*x[4] + x[3]^2 + x[6]

...

MAPLE

with(Groebner):

A263633_row := proc(n) local EE, t1, t2, Q, F, X, p, L, q, c, r;

EE := add(x[i]*t^i, i=1..2*n);

t1 := 1/(1-EE):

t2 := series(t1, t, 2*n):

Q := k -> expand(coeff(t2, t, k));

X := seq(x[i], i=1..n);

p := Q(n);

L := [];

while p <> 0 do

   r := LeadingTerm(p, grlex(X));

   c := r[1]; q := r[2];

   p := p - c*q;

   L := [op(L), c];

od;

L end:

for n from 1 to 8 do A263633_row(n) od; # Program expanded by Peter Luschny, Sep 26 2016

CROSSREFS

For triangle of coefficients of exponential Bell polynomials see A178867.

Cf. A000041, A263634.

Cf. A007318, A133314, A133932.

Sequence in context: A228350 A220482 A084580 * A171850 A087782 A296774

Adjacent sequences:  A263630 A263631 A263632 * A263634 A263635 A263636

KEYWORD

nonn,tabf

AUTHOR

N. J. A. Sloane, Oct 28 2015

EXTENSIONS

More terms and some edits by Peter Luschny, Sep 26 2016

STATUS

approved

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Last modified April 6 18:38 EDT 2020. Contains 333286 sequences. (Running on oeis4.)