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%I #75 Feb 21 2024 11:22:46
%S 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,
%T 85,120,155,185,205,212,205,185,155,120,85,54,30,14,5,1,1,6,20,50,104,
%U 190,314,478,679,908,1151,1390,1605,1776,1886,1924,1886,1776
%N 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).
%C 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.
%C Row lengths are 1, 3, 7, 13, 21, 31, 43, 57, ...: see A002061. - _Michel Marcus_, May 17 2013
%C 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
%C 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
%D 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).
%D J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
%H Jean-François Alcover, <a href="/A162206/b162206.txt">Table of n, a(n) for n = 1..9020 [30 rows]</a>
%H Marwa Ben Abdelmaksoud and Adel Hamdi, <a href="https://arxiv.org/abs/1912.08551">width-k Eulerian polynomials of type A and B and its Gamma-positivity</a>, arXiv:1912.08551 [math.CO], 2019.
%H M. Gaichenkov, <a href="https://mathoverflow.net/questions/336756/the-growth-of-maximum-elements-for-the-reflection-group-d-n">The growth of maximum elements for the reflection group $D_n$</a>, MathOverflow, 2019.
%H Thomas Kahle and Christian Stump, <a href="https://arxiv.org/abs/1802.01389">Counting inversions and descents of random elements in finite Coxeter groups</a>, arXiv:1802.01389 [math.CO], 2018-2019.
%H M. Rubey, <a href="http://www.findstat.org/StatisticsDatabase/St001443">St001443: Finite Cartan types ⟶ ℤ</a>, StatisticsDatabase, 2019.
%H <a href="/index/Gre#GROWTH">Index entries for growth series for groups</a>
%e Triangle begins:
%e 1;
%e 1, 2, 1;
%e 1, 3, 5, 6, 5, 3, 1;
%e 1, 4, 9, 16, 23, 28, 30, 28, 23, 16, 9, 4, 1;
%e 1, 5, 14, 30, 54, 85, 120, 155, 185, 205, 212, 205, 185, 155, 120, 85, 54, 30, 14, 5, 1;
%e 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;
%e 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;
%p # Growth series for D_k, truncated to terms of order M. - _N. J. A. Sloane_, Aug 07 2021
%p f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
%p g := proc(k,M) local a,i; global f;
%p a:=f(k)*mul(f(2*i),i=1..k-1);
%p seriestolist(series(a,x,M+1));
%p end proc;
%t 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 T[6] // Flatten (* _Jean-François Alcover_, Mar 25 2020, after PARI *)
%t 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 *)
%o (PARI) {row(n) = Vec(prod(k=1 ,n-1, 1-x^(2*k))*(1-x^n)/(1-x)^n)}; /* _Michael Somos_, Aug 06 2021 */
%Y 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.
%Y Cf. A002061 (row length), A002866 (row sum), A333508 (central coefficient).
%K nonn,tabf
%O 1,3
%A _John Cannon_ and _N. J. A. Sloane_, Nov 30 2009
%E Revised by _N. J. A. Sloane_, Jan 10 2016
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