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

Author

Peter Luschny, May 31 2020

Keywords

• nonn
• tabf

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.)

Examples

			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]
		

Crossrefs

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

Programs

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);