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.
FORMULA
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
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Aug 22 2010
STATUS
approved