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%I #37 Sep 08 2022 08:44:39
%S 1,157,26690,4508570,761974851,128773405047,21762709934980,
%T 3677897920745140,621564749363392901,105044442632566365137,
%U 17752510805031727164870,3000174326048697741925710,507029461102251552321630151,85687978926280231101185088427
%N Gaussian binomial coefficient [ n,2 ] for q = -13.
%D 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.
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
%D M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H Vincenzo Librandi, <a href="/A015265/b015265.txt">Table of n, a(n) for n = 2..200</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (157, 2041, -2197).
%F G.f.: x^2/((1-x)*(1+13*x)*(1-169*x)). - _Ralf Stephan_, Apr 01 2004
%F a(2) = 1, a(3) = 157, a(4) = 26690, a(n) = 157*a(n-1) + 2041*a(n-2) - 2197*a(n-3). - _Vincenzo Librandi_, Oct 28 2012
%F a(n) = (1/2352)*( (1 - (-13)^n)*((-13)^(n-1) - 1) ). - _M. F. Hasler_, Nov 03 2012
%t Table[QBinomial[n, 2, -13], {n, 2, 20}] (* _Vincenzo Librandi_, Oct 28 2012 *)
%o (Sage) [gaussian_binomial(n,2,-13) for n in range(2,14)] # _Zerinvary Lajos_, May 27 2009
%o (Magma) I:=[1,157,26690]; [n le 3 select I[n] else 157*Self(n-1)+2041*Self(n-2)-2197*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Oct 28 2012
%o (PARI) A015265(n,q=-13)=(1-q^n)*(q^(n-1)-1)/2352 \\ _M. F. Hasler_, Nov 03 2012
%Y Cf. Gaussian binomial coefficients [n,2] for q=-2,...,-12: A015249, A015251, A015253, A015255, A015257 A015258, A015259, A015260, A015261, A015262, A015264.
%Y Cf. Gaussian binomial coefficients [n,r] for q=-13: A015286 (r=3), 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
%K nonn,easy
%O 2,2
%A _Olivier Gérard_, Dec 11 1999
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