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A162206 Triangle read by rows in which row n (n>=1) gives coefficients in expansion of the polynomial f(n) * Product( f(2i), i=1..n-1 )/ f(1)^n, where f(k) = 1-x^k. 47
1, 1, 2, 1, 1, 3, 5, 6, 5, 3, 1, 1, 4, 9, 16, 23, 28, 30, 28, 23, 16, 9, 4, 1, 1, 5, 14, 30, 54, 85, 120, 155, 185, 205, 212, 205, 185, 155, 120, 85, 54, 30, 14, 5, 1, 1, 6, 20, 50, 104, 190, 314, 478, 679, 908, 1151, 1390, 1605, 1776, 1886, 1924, 1886, 1776 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

For n >= 3, this polynomial is the Poincaré polynomial (or growth series) for the reflection group (or Weyl group, or finite Coxeter group) D_n.

Row lengths are 1, 3, 7, 13, 21, 31, 43, 57, ... : see A002061. - Michel Marcus, May 17 2013

The asymptotic growth of maximum elements for the reflection group D_n is about 2(n-1/2) (compare with A000140). - Mikhail Gaichenkov, Aug 21 2019

REFERENCES

N. Bourbaki, Groups et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10a, page 231, W(t).

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

LINKS

Table of n, a(n) for n=1..63.

M. Gaichenkov, The growth of maximum elements for the reflection group $D_n$, MathOverflow, 2019.

M. Rubey, St001443: Finite Cartan types ⟶ ℤ, StatisticsDatabase, 2019.

EXAMPLE

Triangle begins:

1

1,2,1,

1,3,5,6,5,3,1,

1,4,9,16,23,28,30,28,23,16,9,4,1,

1,5,14,30,54,85,120,155,185,205,212,205,185,155,120,85,54,30,14,5,1,

1,6,20,50,104,190,314,478,679,908,1151,1390,1605,1776,1886,1924,1886,1776,1605,1390,1151,908,679,478,314,190,104,50,20,6,1,

1,7,27,77,181,371,686,1169,1862,2800,4005,5481,7210,9149,11230,13363,15442,17353,18983,20230,21013,21280,21013,20230,18983,17353,15442,13363,11230,9149,7210,5481,4005,2800,1862,1169,686,371,181,77,27,7,1,

PROG

(PARI) tabl(nn) = {for (n=1, nn, x = y+y*O(y^(n^2)); v = Vec((1-x^n)*prod(k=1, n-1, 1-x^(2*k))/(1-x)^n); for (i=1, #v, if (v[i], print1(v[i], ", ")); ); print(); ); } \\ Michel Marcus, May 17 2013

CROSSREFS

The growth series for D_k, k >= 5, that is, rows 5 through 12 of this triangle, are A162208-A162212, A162248, A162288, A162297.

Cf. A002061.

Sequence in context: A210098 A241188 A145236 * A075248 A128325 A307883

Adjacent sequences:  A162203 A162204 A162205 * A162207 A162208 A162209

KEYWORD

nonn,tabf

AUTHOR

John Cannon and N. J. A. Sloane, Nov 30 2009

EXTENSIONS

Revised by N. J. A. Sloane, Jan 10 2016

STATUS

approved

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Last modified October 22 19:53 EDT 2019. Contains 328319 sequences. (Running on oeis4.)