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%I #24 Dec 07 2019 12:18:18
%S 1,26521,761974851,21752862899691,621305270140974342,
%T 17745052029585350965782,506816536013640476467362442,
%U 14475186854407942097510802411322
%N Gaussian binomial coefficient [ n,4 ] 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="/A015303/b015303.txt">Table of n, a(n) for n = 4..200</a>
%H <a href="/index/Ga#Gaussian_binomial_coefficients">Index entries related to Gaussian binomial coefficients</a>.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (26521,58611410,-9905328290,-128011801489,137858491849).
%F a(n) = product_{i=1..4} ((-13)^(n-i+1)-1)/((-13)^i-1). - _M. F. Hasler_, Nov 03 2012
%F G.f.: -x^4 / ( (x-1)*(169*x-1)*(2197*x+1)*(13*x+1)*(28561*x-1) ). - _R. J. Mathar_, Aug 03 2016
%e To illustrate the relation qC(n,r)=qC(n,n-r), here with r=4, n=r+1...r+3:
%e A015303(5) = 26521 = A015000(5),
%e A015303(6) = 761974851 = A015265(6),
%e A015303(7) = 21752862899691 = A015286(7).
%t Table[QBinomial[n, 4, -13], {n, 4, 20}] (* _Vincenzo Librandi_, Oct 29 2012 *)
%o (Sage) [gaussian_binomial(n,4,-13) for n in range(4,12)] # _Zerinvary Lajos_, May 27 2009
%o (PARI) A015303(n,r=4,q=-13)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ _M. F. Hasler_, Nov 03 2012
%Y Cf. q-integers and Gaussian binomial coefficients [n,r] for q=-13: A015000, A015265 (r=2), A015286 (r=3), 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
%Y Fifth row (r=4) or column (resp. diagonal) of A015129, read as square (resp. triangular) array.
%K nonn,easy
%O 4,2
%A _Olivier Gérard_, Dec 11 1999
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