A128084 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.

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, 1, 5, 14, 30, 54, 86, 125, 169, 215, 259, 297, 325, 340, 340, 325, 297, 259, 215, 169, 125, 86, 54, 30, 14, 5, 1, 1, 6, 20, 50, 104, 190, 315, 484, 699, 958, 1255, 1580, 1919

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

Cf. A000165 ((2n)!!); A128085 (central terms); A128086 (diagonal), A128087 (row squared sums); A128080, A002522 (row lengths).

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.

Sequence in context: A152067 A286756 A193884 * A131823 A089722 A172356

Adjacent sequences: A128081 A128082 A128083 * A128085 A128086 A128087

Keyword

nonn,tabf

Author

Paul D. Hanna, Feb 14 2007

Status

approved