A006103 Gaussian binomial coefficient [ 2n,n ] for q=3.

1, 4, 130, 33880, 75913222, 1506472167928, 267598665689058580, 427028776969176679964080, 6129263888495201102915629695046, 791614563787525746761491781638123230424, 920094266641283414155073889843358388073398779900

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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..25
• M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

Crossrefs

Cf. A022167.

Programs

Magma

q:=3; [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, 3], {n, 0, 10}] (* Vladimir Reshetnikov, Sep 12 2016 *)

PARI

q=3; {a(n) = prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1))) };
vector(15, n, n--; a(n)) \\ G. C. Greubel, Mar 09 2019

Sage

[gaussian_binomial(2*n,n,3) for n in (0..15)] # G. C. Greubel, Mar 09 2019

Formula

a(n) = Sum_{k=0..n} 3^(k^2)*(A022167(n,k))^2. - Werner Schulte, Mar 09 2019