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

Author

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

Keywords

• nonn
• tabf

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

Examples

			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;
		

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

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).

Programs

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 *)

PARI

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

Extensions

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