A335311 Coefficients of polynomials arising in the series expansion of the multiplicative inverse of an analytic function. Irregular triangle read by rows.

1, 1, 2, 2, 6, 12, 3, 24, 72, 24, 24, 4, 120, 480, 180, 360, 40, 120, 5, 720, 3600, 1440, 4320, 360, 2160, 720, 60, 240, 180, 6, 5040, 30240, 12600, 50400, 3360, 30240, 20160, 630, 5040, 3780, 7560, 84, 420, 840, 7

Offset

0,3

Comments

The coefficients of Bell-type polynomials where the monomials correspond to integer partitions. The monomials are in graded lexicographic order with variables x[0] > x[1] > ... > x[n]. This means that monomials are compared first by their total degree, with ties broken by lexicographic order. (This is the monomial order of Maple after sorting.)

Links

Table of n, a(n) for n=0..44.

Example

The triangle starts (the refinement is indicated by square brackets):
[0]    1;
[1]    1;
[2]    2,     2;
[3]    6,    12,     3;
[4]   24,    72,    (24,    24),    4;
[5]  120,   480,   (180,   360),   (40,   120),     5;
[6]  720,  3600,  (1440,  4320),  (360,  2160,   720), (60,  240,  180),    6;
[7] 5040, 30240, (12600, 50400), (3360, 30240, 20160), (630, 5040, 3780, 7560), (84, 420, 840), 7;
[8] 40320, 282240, (120960, 604800), (33600, 403200, 403200), (6720, 80640, 60480,
241920, 40320), (1008, 10080, 20160, 20160, 30240), (112, 672, 1680, 1120), 8;
The multivariate polynomials start:
        1
        x[0]
      2*x[0]^2 +          2*x[1]
      6*x[0]^3 +    12*x[0]*x[1] +          3*x[2]
     24*x[0]^4 +  72*x[0]^2*x[1] +    24*x[0]*x[2] +       24*x[1]^2 +       4*x[3]
    120*x[0]^5 + 480*x[0]^3*x[1] + 180*x[0]^2*x[2] + 360*x[0]*x[1]^2 + 40*x[0]*x[3] + 120*x[1]*x[2] + 5*x[4]

Maple

A335311Triangle := proc(numrows) local ser, p, C, B, P;
B(0) := 1; ser := series(1/B(s), s, numrows);
C := [seq(expand(simplify(n!*coeff(ser, s, n))), n=0..numrows-1)]:
P := subs(seq((D@@n)(B)(0)=n*x[n], n=1..numrows), C):
for p in P do print(seq(abs(c), c=coeffs(sort(p)))) od end:
A335311Triangle(8);

Crossrefs

Cf. A199673 (row reversed refinement), A006153 (row sums), A000041 (length of rows), A182779 (different monomial order).

Sequence in context: A275312 A209026 A091764 * A192933 A079005 A281351

Adjacent sequences: A335308 A335309 A335310 * A335312 A335313 A335314

Keyword

nonn,tabf

Author

Peter Luschny, May 31 2020

Status

approved