cp's OEIS Frontend

A130810 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).

%I A130810 #51 Jul 25 2025 06:11:55
%S A130810 16,80,240,560,1120,2016,3360,5280,7920,11440,16016,21840,29120,38080,
%T A130810 48960,62016,77520,95760,117040,141680,170016,202400,239200,280800,
%U A130810 327600,380016,438480,503440,575360,654720,742016,837760,942480,1056720
%N A130810 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 A130810 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 A130810 a(n) is the number of 3-dimensional elements in an n-cross polytope where n>=4. - _Patrick J. McNab_, Jul 06 2015
%H A130810 Harlan J. Brothers, <a href="http://www.brotherstechnology.com/docs/Pascal&#39;s_Prism_(supplement).pdf">Pascal's Prism: Supplementary Material</a>.
%H A130810 Milan Janjić, <a href="https://web.archive.org/web/20101104210723/http://www.pmfbl.org/janjic/enumfor.pdf">Two Enumerative Functions</a>. [Wayback Machine link]
%H A130810 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CrossPolytope.html">Cross Polytope</a>.
%F A130810 a(n) = binomial(2*n,4) + binomial(n,2) - n*binomial(2*n-2,2).
%F A130810 a(n) = binomial(n,4)*16. - _Zerinvary Lajos_, Dec 07 2007
%F A130810 G.f.: 16*x^4/(1-x)^5. - _Colin Barker_, Apr 14 2012
%F A130810 a(n) = 2*n*(n-1)*(n-2)*(n-3)/3 = 2*A162668(n-3). - _Robert Israel_, Jul 06 2015
%F A130810 a(n) = 16 * A000332(n). - _Alois P. Heinz_, Oct 26 2020
%F A130810 E.g.f.: 2*exp(x)*x^4/3. - _Stefano Spezia_, Jul 17 2025
%F A130810 From _Amiram Eldar_, Jul 25 2025: (Start)
%F A130810 Sum_{n>=4} 1/a(n) = 1/12.
%F A130810 Sum_{n>=4} (-1)^n/a(n) = 2*log(2) - 4/3. (End)
%p A130810 a:= n-> binomial(2*n,4) +binomial(n,2) -n*binomial(2*n-2,2);
%p A130810 seq(binomial(n, n-4)*2^4, n=4..37); # _Zerinvary Lajos_, Dec 07 2007
%t A130810 a[n_] := 16 * Binomial[n, 4]; Array[a, 34, 4] (* _Amiram Eldar_, Jul 25 2025 *)
%o A130810 (PARI) a(n) = 16 * binomial(n, 4); \\ _Amiram Eldar_, Jul 25 2025
%Y A130810 Cf. A000332, A038207, A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A046092, A130809.  Equals twice A162668.
%K A130810 nonn,easy
%O A130810 4,1
%A A130810 _Milan Janjic_, Jul 16 2007