A286721 Column k=2 of the triangle A286718; Sheffer ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)).

1, 12, 159, 2485, 45474, 959070, 22963996, 616224492, 18331744896, 599061555136, 21339235262784, 823098817664448, 34183157124707200, 1520908498941532800, 72182781516370886400, 3640264913563748243200, 194408478299496756556800, 10961007293837647131724800

Offset

0,2

Comments

a(n) is, for n >= 1, the total volume of the binomial(n+2, n) rectangular polytopes (hyper-cuboids) built from n orthogonal vectors with lengths of the sides from the set {1 + 3*j | j=0..n+1}. See the formula a(n) = sigma[3,1]^{(n+2)}_n and an example below.

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = A286718(n+2, 2), n >= 0.

E.g.f.: (d^2/dx^2)((1 - 3*x)^(-1/3)*((-1/3)*log(1 - 3*x))^2/2!) = (2*(log(1-3*x))^2 - 15*log(1-3*x) + 9)/(3^2*(1-3*x)^(7/3)).

a(n) = sigma[3,1]^{(n+2)}_n, n >= 0, with the elementary symmetric function sigma[3,1]^{n+2}_n of degree n of the n+2 numbers 1, 4, 7, ..., (1 + 3*(n+1)).

Example

a(2) = 159 because sigma[3,1]^{(4)}_2 = 1*(4 + 7 + 10) + 4*(7 + 10) + 7*10 = 159. There are six rectangles (2D rectangular polytopes) built from two orthogonal vectors of different lengths from the set of {1,4,7,10} of total area 159.

Crossrefs

Cf. A007559 (k=0), A024216 (k=1), A286718.

Sequence in context: A269411 A347748 A226297 * A349154 A112719 A091019

Adjacent sequences: A286718 A286719 A286720 * A286722 A286723 A286724

Keyword

nonn,easy

Author

Wolfdieter Lang, May 29 2017

Status

approved