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

0,3

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.

Links

Table of n, a(n) for n=0..23.

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)

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 *)

Crossrefs

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

Sequence in context: A218778 A320521 A371605 * A355352 A015945 A015954

Adjacent sequences: A180263 A180264 A180265 * A180267 A180268 A180269

Keyword

nonn

Author

Robert G. Wilson v, Aug 22 2010

Status

approved