%I #28 Sep 08 2022 08:45:30

%S 128,1024,4608,15360,42240,101376,219648,439296,823680,1464320,

%T 2489344,4073472,6449664,9922560,14883840,21829632,31380096,44301312,

%U 61529600,84198400,113667840,151557120,199779840,260582400,336585600,430829568

%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 7-subsets of X containing none of X_i, (i=1,...n).

%C Number of n permutations (n>=7) of 3 objects u,v,z, with repetition allowed, containing n-7 u's. Example: if n=7 then n-7 =(0) zero u, a(1)=128. - _Zerinvary Lajos_, Aug 05 2008

%C a(n) is the number of 6-dimensional elements in an n-cross polytope where n>=7. - _Patrick J. McNab_, Jul 06 2015

%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,7) + binomial(n,2)*binomial(2*n-4,3) - n*binomial(2*n-2,5) - (2*n-6)*binomial(n,3).

%F a(n) = C(n,n-7)*2^7, n>=7. - _Zerinvary Lajos_, Dec 07 2007

%F G.f.: 128*x^7/(1-x)^8. - _Colin Barker_, Mar 18 2012

%F a(n) = 128*A000580(n). a(n+1) = 2*(n+1)*a(n)/(n-6) for n >= 7. - _Robert Israel_, Jul 08 2015

%p a:=n->binomial(2*n,7)+binomial(n,2)*binomial(2*n-4,3)-n*binomial(2*n-2,5)-(2*n-6)*binomial(n,3);

%p seq(binomial(n,n-7)*2^7,n=7..32); # _Zerinvary Lajos_, Dec 07 2007

%p seq(binomial(n+6, 7)*2^7, n=1..22); # _Zerinvary Lajos_, Aug 05 2008

%t Table[Binomial[n, n - 7] 2^7, {n, 7, 40}] (* _Vincenzo Librandi_, Jul 09 2015 *)

%o (Magma) [Binomial(n,n-7)*2^7: n in [7..40]]; // _Vincenzo Librandi_, Jul 09 2015

%Y Cf. A038207, A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A046092, A130809, A130810, A130811, A130812. - _Zerinvary Lajos_, Aug 05 2008

%Y Cf. A000580.

%K nonn

%O 7,1

%A _Milan Janjic_, Jul 16 2007