A335311 - OEIS

%I #19 Jun 01 2020 06:48:58

%S 1,1,2,2,6,12,3,24,72,24,24,4,120,480,180,360,40,120,5,720,3600,1440,

%T 4320,360,2160,720,60,240,180,6,5040,30240,12600,50400,3360,30240,

%U 20160,630,5040,3780,7560,84,420,840,7

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

%C 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.)

%e The triangle starts (the refinement is indicated by square brackets):

%e [0]    1;

%e [1]    1;

%e [2]    2,     2;

%e [3]    6,    12,     3;

%e [4]   24,    72,    (24,    24),    4;

%e [5]  120,   480,   (180,   360),   (40,   120),     5;

%e [6]  720,  3600,  (1440,  4320),  (360,  2160,   720), (60,  240,  180),    6;

%e [7] 5040, 30240, (12600, 50400), (3360, 30240, 20160), (630, 5040, 3780, 7560), (84, 420, 840), 7;

%e [8] 40320, 282240, (120960, 604800), (33600, 403200, 403200), (6720, 80640, 60480,

%e 241920, 40320), (1008, 10080, 20160, 20160, 30240), (112, 672, 1680, 1120), 8;

%e The multivariate polynomials start:

%e         1

%e         x[0]

%e       2*x[0]^2 +          2*x[1]

%e       6*x[0]^3 +    12*x[0]*x[1] +          3*x[2]

%e      24*x[0]^4 +  72*x[0]^2*x[1] +    24*x[0]*x[2] +       24*x[1]^2 +       4*x[3]

%e     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]

%p A335311Triangle := proc(numrows) local ser, p, C, B, P;

%p B(0) := 1; ser := series(1/B(s), s, numrows);

%p C := [seq(expand(simplify(n!*coeff(ser,s,n))), n=0..numrows-1)]:

%p P := subs(seq((D@@n)(B)(0)=n*x[n], n=1..numrows), C):

%p for p in P do print(seq(abs(c), c=coeffs(sort(p)))) od end:

%p A335311Triangle(8);

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

%K nonn,tabf

%O 0,3

%A _Peter Luschny_, May 31 2020