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%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
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