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A130813 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).

%I A130813 #29 Feb 16 2025 08:33:06
%S A130813 128,1024,4608,15360,42240,101376,219648,439296,823680,1464320,
%T A130813 2489344,4073472,6449664,9922560,14883840,21829632,31380096,44301312,
%U A130813 61529600,84198400,113667840,151557120,199779840,260582400,336585600,430829568
%N A130813 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 A130813 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 A130813 a(n) is the number of 6-dimensional elements in an n-cross polytope where n>=7. - _Patrick J. McNab_, Jul 06 2015
%H A130813 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A130813 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CrossPolytope.html">Cross Polytope</a>
%F A130813 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 A130813 a(n) = C(n,n-7)*2^7, n>=7. - _Zerinvary Lajos_, Dec 07 2007
%F A130813 G.f.: 128*x^7/(1-x)^8. - _Colin Barker_, Mar 18 2012
%F A130813 a(n) = 128*A000580(n). a(n+1) = 2*(n+1)*a(n)/(n-6) for n >= 7. - _Robert Israel_, Jul 08 2015
%p A130813 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 A130813 seq(binomial(n,n-7)*2^7,n=7..32); # _Zerinvary Lajos_, Dec 07 2007
%p A130813 seq(binomial(n+6, 7)*2^7, n=1..22); # _Zerinvary Lajos_, Aug 05 2008
%t A130813 Table[Binomial[n, n - 7] 2^7, {n, 7, 40}] (* _Vincenzo Librandi_, Jul 09 2015 *)
%o A130813 (Magma) [Binomial(n,n-7)*2^7: n in [7..40]]; // _Vincenzo Librandi_, Jul 09 2015
%Y A130813 Cf. A038207, A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A046092, A130809, A130810, A130811, A130812. - _Zerinvary Lajos_, Aug 05 2008
%Y A130813 Cf. A000580.
%K A130813 nonn
%O A130813 7,1
%A A130813 _Milan Janjic_, Jul 16 2007