A180266 a(0) = 0; a(n) = C(2*n-2,n-1)*(n^2-2*n+2)/n for n >= 1.

0, 1, 2, 10, 50, 238, 1092, 4884, 21450, 92950, 398684, 1696396, 7171892, 30161740, 126293000, 526864680, 2191034970, 9086921190, 37596989100, 155232577500, 639749274780, 2632212288420, 10814090022840, 44369043365400

Offset: 0

Author

Robert G. Wilson v, Aug 22 2010

Keywords

• nonn

Comments

We may define Figurate Numbers F(r,n,d) with rank r, index n in dimension d as F(r,n,d) = binomial(r+d-2,d-1) *((r-1)*(n-2)+d) /d. These are polygonal numbers A057145 or A086271 in d=2, pyramidal numbers A080851 in d=3, and 4D pyramidal numbers A080852 in d=4, for example.

This sequence here is a(n) = F(n,n,n), the n-th n-gonal figurate number in n dimensions.

Limit_{n -> infinity} a(n+1)/a(n) = 4. - Robert G. Wilson v, Oct 30 2013

References

• Albert H. Beiler, Recreations in the Theory of Numbers, The Queen of Mathematics Entertains, Second Edition, Dover, New York, 1966, Chptr. XVIII Ball Games, p. 196.

Crossrefs

Cf. A000984, A024483, A057145, A080851, A080852, A086271.

Programs

Mathematica

Figurate[ngon_, rank_, dim_] := Binomial[rank + dim - 2, dim - 1] ((rank - 1)*(ngon - 2) + dim)/dim; Table[ Figurate[n, n, n], {n, 50}]
Join[{0},Table[Binomial[2n-2,n-1] (n^2-2n+2)/n,{n,30}]] (* Harvey P. Dale, Sep 22 2019 *)

Formula

a(n) = A000984(n-1) + (n-1)*A024483(n). [R. J. Mathar, Nov 18 2010]

From Ilya Gutkovskiy, Mar 29 2018: (Start)

O.g.f.: 1 - (1 - 7*x + 10*x^2)/(1 - 4*x)^(3/2).

E.g.f.: 1 - exp(2*x)*((1 - 3*x)*BesselI(0,2*x) + 2*x*BesselI(1,2*x)).

a(n) = [x^n] x*(1 - 3*x + n*x)/(1 - x)^(n+1). (End)