A162206 Triangle read by rows in which row n (n >= 1) gives coefficients in expansion of the polynomial f(n) * Product_{i=1..n-1} f(2i), where f(k) = (1 - x^k)/(1-x).

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

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

Row maxima ~ 2^(n-1)*n!/(sigma * sqrt(3/Pi)), sigma^2 = (4*n^3 - 3*n^2 - n)/36 = variance of D_n. - Mikhail Gaichenkov, Feb 08 2023

References

N. Bourbaki, Groupes 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

Jean-François Alcover, Table of n, a(n) for n = 1..9020 [30 rows]

Marwa Ben Abdelmaksoud and Adel Hamdi, width-k Eulerian polynomials of type A and B and its Gamma-positivity, arXiv:1912.08551 [math.CO], 2019.

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

Thomas Kahle and Christian Stump, Counting inversions and descents of random elements in finite Coxeter groups, arXiv:1802.01389 [math.CO], 2018-2019.

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

Index entries for growth series for groups

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;

Maple

# Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k, M) local a, i; global f;
a:=f(k)*mul(f(2*i), i=1..k-1);
seriestolist(series(a, x, M+1));
end proc;

Mathematica

T[nn_] := Reap[Do[x = y + y O[y]^(n^2); v = (1 - x^n) Product[1 - x^(2k), {k, 1, n - 1}]/(1 - x)^n // CoefficientList[#, y]&; Sow[v], {n, nn}]][[2, 1]];
T[6] // Flatten (* Jean-François Alcover, Mar 25 2020, after PARI *)
T[ n_] := Module[{x}, CoefficientList[ Product[1 - x^(2 k), {k, 1, n - 1}] (1 - x^n) /(1 - x)^n // Expand, x]] (* Michael Somos, Aug 06 2021 *)

Prog

(PARI) {row(n) = Vec(prod(k=1 , n-1, 1-x^(2*k))*(1-x^n)/(1-x)^n)}; /* Michael Somos, Aug 06 2021 */

Crossrefs

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

Cf. A002061 (row length), A002866 (row sum), A333508 (central coefficient).

Sequence in context: A210098 A241188 A145236 * A075248 A359140 A365623

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