A015286 Gaussian binomial coefficient [ n,3 ] for q = -13.
1, -2040, 4508570, -9900819720, 21752862899691, -47790911017216080, 104996653267533662740, -230677643550873536294640, 506798783502833908602716981, -1113436927250681654567602842120
Offset: 3
Examples
A015286(7) = 21752862899691 = A015303(7), A015286(8) = -47790911017216080 = A015321(8), A015286(9) = 104996653267533662740 = A015337(9). - _M. F. Hasler_, Nov 03 2012
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 = 3..200
- Index entries related to Gaussian binomial coefficients.
- Index entries for linear recurrences with constant coefficients, signature (-2040,346970,4481880,-4826809)
Crossrefs
Cf. Gaussian binomial coefficients [n,r] for q=-13: A015265 (r=2), A015303 (r=4), A015321 (r=5), A015337 (r=6), A015355 (r=7), A015370 (r=8), A015385 (r=9), A015402 (r=10), A015422 (r=11), A015438 (r=12). - M. F. Hasler, Nov 03 2012
Fourth row (r=3) or column (resp. diagonal) in A015129 (read as square array resp. triangle). - M. F. Hasler, Nov 03 2012
Programs
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Magma
r:=3; q:=-13; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 02 2016
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Mathematica
QBinomial[Range[3,15],3,-13] (* Harvey P. Dale, Jun 21 2012 *) Table[QBinomial[n, 3, -13], {n, 3, 20}] (* Vincenzo Librandi, Oct 28 2012 *)
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PARI
A015286(n,r=3,q=-13)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012
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Sage
[gaussian_binomial(n,3,-13) for n in range(3,13)] # Zerinvary Lajos, May 27 2009
Formula
a(n) = Product_{i=1..3} ((-13)^(n-i+1) - 1)/((-13)^i - 1). - M. F. Hasler, Nov 03 2012
G.f.: x^3 / ( (x-1)*(2197*x+1)*(13*x+1)*(169*x-1) ). - R. J. Mathar, Aug 03 2016