|
%I M5479 #31 Sep 08 2022 08:44:34
%S 1,781,508431,320327931,200525284806,125368356709806,
%T 78360229974772306,48975769621072897306,30609934249224268600431,
%U 19131218685276848401412931,11957012900737114492991256681,7473133215765585192791624069181,4670708278954101902438990598678556
%N Gaussian binomial coefficient [ n,4 ] for q = 5.
%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 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H Vincenzo Librandi, <a href="/A006113/b006113.txt">Table of n, a(n) for n = 4..200</a>
%H M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
%F G.f.: x^4/((1-x)*(1-5*x)*(1-25*x)*(1-125*x)*(1-625*x)). - _Vincenzo Librandi_, Aug 07 2016
%F a(n) = Product_{i=1..4} (5^(n-i+1)-1)/(5^i-1), by definition. - _Vincenzo Librandi_, Aug 06 2016
%p qBinom := proc(n,m,q)
%p mul( (1-q^(n-i))/(1-q^(i+1)),i=0..m-1) ;
%p end proc:
%p A006113 := proc(n)
%p qBinom(n,4,5) ;
%p end proc:
%p seq(A006113(n),n=4..16) ; # _R. J. Mathar_, Sep 28 2011
%t Table[QBinomial[n, 4, 5], {n, 4, 20}] (* _Vincenzo Librandi_, Aug 07 2016 *)
%o (Sage) [gaussian_binomial(n,4,5) for n in range(4,14)] # _Zerinvary Lajos_, May 27 2009
%o (Magma) r:=4; q:=5; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // _Vincenzo Librandi_, Aug 07 2016
%K nonn,easy
%O 4,2
%A _N. J. A. Sloane_
|
