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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. 0
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, May 31 2020
STATUS
approved

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Last modified March 28 07:48 EDT 2024. Contains 371235 sequences. (Running on oeis4.)