A182779 -id:A182779 - cp's OEIS frontend

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

Peter Luschny, May 31 2020

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.)

			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]

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

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);

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

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