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A006103 Gaussian binomial coefficient [ 2n,n ] for q=3.
(Formerly M3715)
1
1, 4, 130, 33880, 75913222, 1506472167928, 267598665689058580, 427028776969176679964080, 6129263888495201102915629695046, 791614563787525746761491781638123230424, 920094266641283414155073889843358388073398779900 (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

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)

FORMULA

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

MATHEMATICA

Table[QBinomial[2n, n, 3], {n, 0, 10}] (* Vladimir Reshetnikov, Sep 12 2016 *)

PROG

(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

(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

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

CROSSREFS

Cf. A022167.

Sequence in context: A096759 A299367 A299931 * A209012 A003371 A113253

Adjacent sequences:  A006100 A006101 A006102 * A006104 A006105 A006106

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 23 12:43 EDT 2019. Contains 321430 sequences. (Running on oeis4.)