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%I #12 Dec 19 2017 02:15:52
%S 1,15,231,4040,80844,1835988,46819324,1327098024,41436870696,
%T 1414064576672,52383613213920,2094099207620160,89873259151044160,
%U 4122137910567440640,201246677825480820480,10420702442559832716800,570477791902749185318400,32923432900388514432614400
%N Column k=2 of the triangle A225470; Sheffer ((1 - 3*x)^(-2/3), (-1/3)*log(1 - 3*x)).
%C 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.
%F a(n) = A225470(n+2, 2), n >= 0.
%F 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)).
%F 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)).
%e 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.
%Y Cf. A008544 (k=0), A024395 (k=1), A225470.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, May 29 2017
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