A015386 Gaussian binomial coefficient [ n,10 ] for q=-2.

1, 683, 932295, 848699215, 926949282623, 920460637644639, 957498220445101855, 972884994173649887135, 1000137219716325891620511, 1022146087305755916943130783, 1047699739488399814866709052575, 1072321450350081081965428740719775

Offset: 10

Author

Olivier Gérard

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 10..200
• Index entries for linear recurrences with constant coefficients, signature (683,465806,-106203768, -14443712448,903388560384,28908433932288,-473291569496064, -3563607111499776,16004972290244608,24030926136672256,-36028797018963968).

Crossrefs

Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13: A015388, A015390, A015391, A015392, A015393, A015394, A015397, A015398, A015399, A015401, A015402.

Programs

Magma

r:=10; q:=-2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 04 2012

Mathematica

Table[QBinomial[n, 10, -2],{n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

Sage

[gaussian_binomial(n,10,-2) for n in range(10,21)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..10} ((-2)^(n-i+1)-1)/((-2)^i-1) (by definition). - Vincenzo Librandi, Nov 04 2012

G.f.: x^10 / ( (x-1)*(512*x+1)*(64*x-1)*(128*x+1)*(1024*x-1)*(2*x+1)*(8*x+1)*(32*x+1)*(16*x-1)*(4*x-1)*(256*x-1) ). - R. J. Mathar, Sep 22 2016