A006098 Gaussian binomial coefficient [ 2n,n ] for q=2. (Formerly M3138)

1, 3, 35, 1395, 200787, 109221651, 230674393235, 1919209135381395, 63379954960524853651, 8339787869494479328087443, 4380990637147598617372537398675, 9196575543360038413217351554014467475, 77184136346814161837268404381760884963259795

Offset

0,2

References

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Links

T. D. Noe, Table of n, a(n) for n = 0..35

Alin Bostan and Sergey Yurkevich, On the q-analogue of Pólya's Theorem, arXiv:2109.02406 [math.CO], 2021.

I. Siap and I. Aydogdu, Counting The Generator Matrices of Z_2 Z_8 Codes, arXiv:1303.6985 [math.CO], 2013.

M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

Formula

a(n) = A022166(2n,n). - Alois P. Heinz, Mar 30 2016

a(n) ~ c * 2^(n^2), where c = A065446. - Vaclav Kotesovec, Sep 22 2016

a(n) = Sum_{k=0..n} 2^(k^2)*(A022166(n,k))^2. - Werner Schulte, Mar 09 2019

Mathematica

Table[QBinomial[2n, n, 2], {n, 0, 20}] (* Harvey P. Dale, Oct 22 2012 *)

Prog

(Sage) [gaussian_binomial(2*n, n, 2) for n in range(0, 11)] # Zerinvary Lajos, May 25 2009
(PARI) q=2; {a(n) = prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1))) };
vector(10, n, n--; a(n)) \\ G. C. Greubel, Mar 09 2019
(Magma) q:=2; [n le 0 select 1 else (&*[(1-q^(2*n-j))/(1-q^(j+1)): j in [0..n-1]]): n in [0..15]]; // G. C. Greubel, Mar 09 2019

Crossrefs

Cf. A022166, A065446.

Sequence in context: A215582 A136525 A136556 * A320845 A012499 A125530

Adjacent sequences: A006095 A006096 A006097 * A006099 A006100 A006101

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Harvey P. Dale, Oct 22 2012

Status

approved