A286722 Column k=2 of the triangle A225470; Sheffer ((1 - 3*x)^(-2/3), (-1/3)*log(1 - 3*x)).

1, 15, 231, 4040, 80844, 1835988, 46819324, 1327098024, 41436870696, 1414064576672, 52383613213920, 2094099207620160, 89873259151044160, 4122137910567440640, 201246677825480820480, 10420702442559832716800, 570477791902749185318400, 32923432900388514432614400

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 {2 + 3*j | j=0..n+1}. See the formula a(n) = sigma[3,2]^{(n+2)}_n and an example below.

Links

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

Formula

a(n) = A225470(n+2, 2), n >= 0.

E.g.f.: (d^2/dx^2)((1 - 3*x)^(-2/3)*((-1/3)*log(1 - 3*x))^2/2!) = (5*(log(1-3*x))^2 - 21*log(1-3*x) + 9)/(3^2*(1-3*x)^(8/3)).

a(n) = sigma[3,2]^{(n+2)}_n, n >= 0, with the elementary symmetric function sigma[3,2]^{n+2}_n of degree n of the n+2 numbers 2, 5, 8, ..., (2 + 3*(n+1)).

Example

a(2) = 231 because sigma[3,2]^{(4)}_2 = 2*(5 + 8 + 11) + 5*(8 + 11) + 8*11 = 231. There are six rectangles (2D rectangular polytopes) built from two orthogonal vectors of different lengths from the set of {2,5,8,11} of total area 231.

Crossrefs

Cf. A008544 (k=0), A024395 (k=1), A225470.

Sequence in context: A041422 A129836 A075262 * A250418 A231411 A097185

Adjacent sequences: A286719 A286720 A286721 * A286723 A286724 A286725

Keyword

nonn,easy

Author

Wolfdieter Lang, May 29 2017

Status

approved