A286723 Column k = 1 of the triangle A225471; Sheffer ((1 - 3*x)^(-3/4), (-1/4)*log(1 - 4*x)).

1, 10, 131, 2196, 45189, 1105182, 31354119, 1012861224, 36717532425, 1476342400050, 65212709985675, 3139386801018300, 163605030141437325, 9176588125543758150, 551225830134140520975, 35305848011347321438800, 2401944921672748059294225, 172980447467901106243829850

Offset: 0

Author

Wolfdieter Lang, May 29 2017

Keywords

• nonn
• easy

Comments

a(n) is, for n >= 1, the total volume of the binomial(n+1, n) rectangular polytopes (hyper-cuboids) built from n orthogonal vectors with lengths of the sides from the set {3 + 4*j | j=0..n}. See the formula a(n) = sigma[4,3]^{(n+1)}_n and an example below.

Examples

			a(2) = 131 because sigma[4,3]^{(3)}_2 = 3*(7 + 11) + 7*11 = 131. There are three rectangles (2D rectangular polytopes) built from two orthogonal vectors of different lengths from the set of {3,7,11} of total area 131.
		

Crossrefs

Cf. A008545 (k=0), A225471.

Formula

a(n) = A225471(n+1, 1), n >= 1.

E.g.f.: (d/dx) ((1 - 4*x)^(-3/4)*((-1/4)*log(1 - 4*x))) = (4 - 3*log(1-4*x)) / (4*(1-4*x)^(7/4)).

a(n) = sigma[4,3]^{(n+1)}_n, n >= 0, with the elementary symmetric function sigma[4,3]^{(n+1)}_n of degree n of the n+1 numbers 3, 7, 11, ..., (1 + 4*n), and sigma[4,3]^{(n+1)}_0 := 1.