OFFSET
0,5
COMMENTS
See A128080 for the triangle of coefficients of q in the q-analog of the odd double factorials.
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
LINKS
Paul D. Hanna, Rows n=0..30 of triangle, in flattened form.
Hasan Arslan, A combinatorial interpretation of Mahonian numbers of type B, arXiv:2404.05099 [math.CO], 2024.
Thomas Kahle and Christian Stump, Counting inversions and descents of random elements in finite Coxeter groups, arXiv:1802.01389 [math.CO], 2018-2019.
Ali Kessouri, Moussa Ahmia, Hasan Arslan, and Salim Mesbahi, Combinatorics of q-Mahonian numbers of type B and log-concavity, arXiv:2408.02424 [math.CO], 2024. See p. 6.
Eric Weisstein's World of Mathematics, q-Factorial.
A. V. Yurkin, On similarity of systems of geometrical and arithmetic triangles, in Mathematics, Computing, Education Conference XIX, 2012.
A. V. Yurkin, New view on the diffraction discovered by Grimaldi and Gaussian beams, arXiv preprint arXiv:1302.6287 [physics.optics], 2013.
A. V. Yurkin, New binomial and new view on light theory, LAP Lambert Academic Publishing, 2013, 78 pages.
EXAMPLE
The row sums form A000165, the even double factorial numbers:
[1, 2, 8, 48, 384, 3840, 46080, 645120, ..., (2n)!!, ...].
Triangle begins:
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, 24, 16, 9, 4, 1;
...
MATHEMATICA
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 *)
PROG
(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) ))}
for(n=0, 8, for(k=0, n^2, print1(T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Paul D. Hanna, Feb 14 2007
STATUS
approved