A130810 - OEIS

%I #35 Oct 26 2020 19:07:03

%S 16,80,240,560,1120,2016,3360,5280,7920,11440,16016,21840,29120,38080,

%T 48960,62016,77520,95760,117040,141680,170016,202400,239200,280800,

%U 327600,380016,438480,503440,575360,654720,742016,837760,942480,1056720

%N If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 4-subsets of X containing none of X_i, (i=1,...,n).

%C Number of n permutations (n>=4) of 3 objects u,v,z, with repetition allowed, containing n-4 u's. Example: if n=4 then n-4 =(0) zero u, a(1)=16 because we have vvvv zzzz vvvz zzzv vvzv zzvz vzvv zvzz zvvv vzzz vvzz zzvv vzvz zvzv zvvz vzzv. - _Zerinvary Lajos_, Aug 05 2008

%C a(n) is the number of 3-dimensional elements in an n-cross polytope where n>=4. - _Patrick J. McNab_, Jul 06 2015

%H H. J. Brothers, <a href="http://www.brotherstechnology.com/docs/Pascal&#39;s_Prism_(supplement).pdf">Pascal's Prism: Supplementary Material</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CrossPolytope.html">Cross Polytope</a>

%F a(n) = binomial(2*n,4) +binomial(n,2) -n*binomial(2*n-2,2).

%F a(n) = binomial(n,4)*16. - _Zerinvary Lajos_, Dec 07 2007

%F G.f.: 16*x^4/(1-x)^5. - _Colin Barker_, Apr 14 2012

%F a(n) = 2*n*(n-1)*(n-2)*(n-3)/3 = 2*A162668(n-3). - _Robert Israel_, Jul 06 2015

%F a(n) = 16 * A000332(n). - _Alois P. Heinz_, Oct 26 2020

%p a:= n-> binomial(2*n,4) +binomial(n,2) -n*binomial(2*n-2,2);

%p seq(binomial(n, n-4)*2^4, n=4..37); # _Zerinvary Lajos_, Dec 07 2007

%Y Cf. A000332, A038207, A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A046092, A130809.  Equals twice A162668.

%K nonn,easy

%O 4,1

%A _Milan Janjic_, Jul 16 2007