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

4,1

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

Number of ternary sequences with exactly four 0's and 1's in total. For example, a(5) = 80 since the strings are (number of permutations in parentheses): 00002 (5), 10002 (20), 11002 (30), 11102 (20), 11112 (5). - Enrique Navarrete, Oct 08 2025

Also the number of tetrahedra in the n-cocktail party graph. - Eric W. Weisstein, Dec 19 2025

Links

Table of n, a(n) for n=4..37.

Harlan J. Brothers, Pascal's Prism: Supplementary Material.

Milan Janjić, Two Enumerative Functions. [Wayback Machine link]

Eric Weisstein's World of Mathematics, Cocktail Party Graph.

Eric Weisstein's World of Mathematics, Cross Polytope.

Eric Weisstein's World of Mathematics, Graph Tetrahedron.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

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)

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Enrique Navarrete, Oct 08 2025

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 *)

Prog

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

Crossrefs

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

Sequence in context: A111732 A271992 A008511 * A212090 A212240 A050468

Adjacent sequences: A130807 A130808 A130809 * A130811 A130812 A130813

Keyword

nonn,easy

Author

Milan Janjic, Jul 16 2007

Status

approved