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A006098 Gaussian binomial coefficient [ 2n,n ] for q=2.
(Formerly M3138)
3
1, 3, 35, 1395, 200787, 109221651, 230674393235, 1919209135381395, 63379954960524853651, 8339787869494479328087443, 4380990637147598617372537398675, 9196575543360038413217351554014467475, 77184136346814161837268404381760884963259795 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A215582 A136525 A136556 * A320845 A012499 A125530
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Harvey P. Dale, Oct 22 2012
STATUS
approved

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Last modified May 20 04:44 EDT 2024. Contains 372703 sequences. (Running on oeis4.)