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%I #71 Apr 10 2024 04:44:11
%S 1,1,1,1,2,2,2,1,1,3,5,7,8,8,7,5,3,1,1,4,9,16,24,32,39,44,46,44,39,32,
%T 24,16,9,4,1,1,5,14,30,54,86,125,169,215,259,297,325,340,340,325,297,
%U 259,215,169,125,86,54,30,14,5,1,1,6,20,50,104,190,315,484,699,958,1255,1580,1919
%N Triangle, read by rows of n^2+1 terms, of coefficients of q in the q-analog of the even double factorials: T(n,k) = [q^k] Product_{j=1..n} (1-q^(2j))/(1-q) for n>0, with T(0,0)=1.
%C See A128080 for the triangle of coefficients of q in the q-analog of the odd double factorials.
%C Row maxima ~ 2^n*n!/(sigma * sqrt(2*Pi)), sigma^2 = (4*n^3 + 6*n^2 - n)/36 = variance of Coxeter group B_n (see also A161858). - _Mikhail Gaichenkov_, Feb 08 2023
%H Paul D. Hanna, <a href="/A128084/b128084.txt">Rows n=0..30 of triangle, in flattened form.</a>
%H Hasan Arslan, <a href="https://arxiv.org/abs/2404.05099">A combinatorial interpretation of Mahonian numbers of type B</a>, arXiv:2404.05099 [math.CO], 2024.
%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>.
%H A. V. Yurkin, <a href="http://www.mce.biophys.msu.ru/eng/archive/abstracts/mce19/sect1138/doc150220/">On similarity of systems of geometrical and arithmetic triangles</a>, in Mathematics, Computing, Education Conference XIX, 2012.
%H A. V. Yurkin, <a href="http://arxiv.org/abs/1302.6287">New view on the diffraction discovered by Grimaldi and Gaussian beams</a>, arXiv preprint arXiv:1302.6287 [physics.optics], 2013.
%H A. V. Yurkin, <a href="https://www.lap-publishing.com/catalog/details//store/gb/book/978-3-659-38404-2/new-binomial-and-new-view-on-light-theory">New binomial and new view on light theory</a>, LAP Lambert Academic Publishing, 2013, 78 pages.
%e The row sums form A000165, the even double factorial numbers:
%e [1, 2, 8, 48, 384, 3840, 46080, 645120, ..., (2n)!!, ...].
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 2, 2, 2, 1;
%e 1, 3, 5, 7, 8, 8, 7, 5, 3, 1;
%e 1, 4, 9, 16, 24, 32, 39, 44, 46, 44, 39, 32, 24, 16, 9, 4, 1;
%e ...
%t t[n_, k_] := If[k < 0 || k > n^2, 0, If[n == 0, 1, Coefficient[ Series[ Product[ (1 - q^(2*j))/(1 - q), {j, 1, n}], {q, 0, n^2}], q, k]]]; Table[t[n, k], {n, 0, 6}, {k, 0, n^2}] // Flatten (* _Jean-François Alcover_, Mar 06 2013, translated from Pari *)
%o (PARI) {T(n,k) = if(k<0||k>n^2,0, if(n==0,1, polcoeff( prod(j=1,n,(1-q^(2*j))/(1-q)), k,q) ))}
%o for(n=0,8,for(k=0,n^2,print1(T(n,k),", "));print(""))
%Y Cf. A000165 ((2n)!!); A128085 (central terms); A128086 (diagonal), A128087 (row squared sums); A128080, A002522 (row lengths).
%Y The growth series for the affine Coxeter groups B_2 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858. These are all rows of A128084.
%K nonn,tabf
%O 0,5
%A _Paul D. Hanna_, Feb 14 2007
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