A128085 Central coefficients of q in the q-analog of the even double factorials: a(n) = [q^([n^2/2])] Product_{j=1..n} (1-q^(2j))/(1-q).

1, 1, 2, 8, 46, 340, 3210, 36336, 484636, 7394458, 127707302, 2454109404, 52091631896, 1207854671388, 30431260261770, 826657521349952, 24114046688034516, 751085176539860458, 24899882719111953556

Offset

0,3

Comments

See A128081 for central coefficients of q in the q-analog of the odd double factorials. Also, A000140 is the central coefficients of q-factorials, giving the maximum number of permutations on n letters having the same number of inversions.

Links

Table of n, a(n) for n=0..18.

Eric Weisstein's World of Mathematics, q-Factorial.

Example

a(n) is the central term of the q-analog of even double factorials,
in which the coefficients of q (triangle A128084) begin:
n=0: (1);
n=1: (1),1;
n=2: 1,2,(2),2,1;
n=3: 1,3,5,7,(8),8,7,5,3,1;
n=4: 1,4,9,16,24,32,39,44,(46),44,39,32,24,16,9,4,1;
n=5: 1,5,14,30,54,86,125,169,215,259,297,325,(340),340,325,297,...;...
The terms enclosed in parenthesis are initial terms of this sequence.

Prog

(PARI) a(n)=if(n==0, 1, polcoeff(prod(k=1, n, (1-q^(2*k))/(1-q)), n^2\2, q))

Crossrefs

Cf. A000165 ((2n)!!); A128084 (triangle), A128086 (diagonal); A128081.

Sequence in context: A111552 A321965 A229559 * A052801 A294784 A180390

Adjacent sequences: A128082 A128083 A128084 * A128086 A128087 A128088

Keyword

nonn

Author

Paul D. Hanna, Feb 14 2007

Status

approved