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

Author

Paul D. Hanna, Feb 14 2007

Keywords

• nonn
• tabf

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

Examples

			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;
 ...
		

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.

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.

Programs

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 *)

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(""))

PARI

row(n)=Vec(prod(j=1, n, (1-x^(2*j))/(1-x))) \\ Andrew Howroyd, Mar 21 2025