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).

16, 80, 240, 560, 1120, 2016, 3360, 5280, 7920, 11440, 16016, 21840, 29120, 38080, 48960, 62016, 77520, 95760, 117040, 141680, 170016, 202400, 239200, 280800, 327600, 380016, 438480, 503440, 575360, 654720, 742016, 837760, 942480, 1056720

Offset: 4

Author

Milan Janjic, Jul 16 2007

Keywords

• nonn
• easy

Comments

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

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

Links

• Harlan J. Brothers, Pascal's Prism: Supplementary Material.
• Milan Janjić, Two Enumerative Functions. [Wayback Machine link]
• Eric Weisstein's World of Mathematics, Cross Polytope.

Crossrefs

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

Programs

Maple

a:= n-> binomial(2*n,4) +binomial(n,2) -n*binomial(2*n-2,2);
seq(binomial(n, n-4)*2^4, n=4..37); # Zerinvary Lajos, Dec 07 2007

Mathematica

a[n_] := 16 * Binomial[n, 4]; Array[a, 34, 4] (* Amiram Eldar, Jul 25 2025 *)

PARI

a(n) = 16 * binomial(n, 4); \\ Amiram Eldar, Jul 25 2025

Formula

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

a(n) = binomial(n,4)*16. - Zerinvary Lajos, Dec 07 2007

G.f.: 16*x^4/(1-x)^5. - Colin Barker, Apr 14 2012

a(n) = 2*n*(n-1)*(n-2)*(n-3)/3 = 2*A162668(n-3). - Robert Israel, Jul 06 2015

a(n) = 16 * A000332(n). - Alois P. Heinz, Oct 26 2020

E.g.f.: 2*exp(x)*x^4/3. - Stefano Spezia, Jul 17 2025

From Amiram Eldar, Jul 25 2025: (Start)

Sum_{n>=4} 1/a(n) = 1/12.

Sum_{n>=4} (-1)^n/a(n) = 2*log(2) - 4/3. (End)