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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_{i=1..n-1} f(2i), where f(k) = (1 - x^k)/(1-x).

%I A162206 #75 Feb 21 2024 11:22:46
%S A162206 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 A162206 85,120,155,185,205,212,205,185,155,120,85,54,30,14,5,1,1,6,20,50,104,
%U A162206 190,314,478,679,908,1151,1390,1605,1776,1886,1924,1886,1776
%N 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).
%C A162206 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 A162206 Row lengths are 1, 3, 7, 13, 21, 31, 43, 57, ...: see A002061. - _Michel Marcus_, May 17 2013
%C A162206 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 A162206 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 A162206 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 A162206 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
%H A162206 Jean-François Alcover, <a href="/A162206/b162206.txt">Table of n, a(n) for n = 1..9020 [30 rows]</a>
%H A162206 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 A162206 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 A162206 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 A162206 M. Rubey, <a href="http://www.findstat.org/StatisticsDatabase/St001443">St001443: Finite Cartan types ⟶ ℤ</a>, StatisticsDatabase, 2019.
%H A162206 <a href="/index/Gre#GROWTH">Index entries for growth series for groups</a>
%e A162206 Triangle begins:
%e A162206 1;
%e A162206 1, 2, 1;
%e A162206 1, 3, 5, 6, 5, 3, 1;
%e A162206 1, 4, 9, 16, 23, 28, 30, 28, 23, 16, 9, 4, 1;
%e A162206 1, 5, 14, 30, 54, 85, 120, 155, 185, 205, 212, 205, 185, 155, 120, 85, 54, 30, 14, 5, 1;
%e A162206 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 A162206 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 A162206 # Growth series for D_k, truncated to terms of order M. - _N. J. A. Sloane_, Aug 07 2021
%p A162206 f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
%p A162206 g := proc(k,M) local a,i; global f;
%p A162206 a:=f(k)*mul(f(2*i),i=1..k-1);
%p A162206 seriestolist(series(a,x,M+1));
%p A162206 end proc;
%t A162206 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 A162206 T[6] // Flatten (* _Jean-François Alcover_, Mar 25 2020, after PARI *)
%t A162206 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 A162206 (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 A162206 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 A162206 Cf. A002061 (row length), A002866 (row sum), A333508 (central coefficient).
%K A162206 nonn,tabf
%O A162206 1,3
%A A162206 _John Cannon_ and _N. J. A. Sloane_, Nov 30 2009
%E A162206 Revised by _N. J. A. Sloane_, Jan 10 2016