A006098 Gaussian binomial coefficient [ 2n,n ] for q=2.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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.

Crossrefs

Cf. A022166, A065446.

Programs

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

Mathematica

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

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

Sage

[gaussian_binomial(2*n,n,2) for n in range(0,11)] # Zerinvary Lajos, May 25 2009

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

Extensions

More terms from Harvey P. Dale, Oct 22 2012