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

%I

%S 0,1,2,10,50,238,1092,4884,21450,92950,398684,1696396,7171892,

%T 30161740,126293000,526864680,2191034970,9086921190,37596989100,

%U 155232577500,639749274780,2632212288420,10814090022840,44369043365400

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

%C 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.

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

%C Limit_{n -> infinity} a(n+1)/a(n) = 4. - _Robert G. Wilson v_, Oct 30 2013

%D 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.

%F a(n) = A000984(n-1) + (n-1)*A024483(n). [_R. J. Mathar_, Nov 18 2010]

%F From _Ilya Gutkovskiy_, Mar 29 2018: (Start)

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

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

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

%t Figurate[ngon_, rank_, dim_] := Binomial[rank + dim - 2, dim - 1] ((rank - 1)*(ngon - 2) + dim)/dim; Table[ Figurate[n, n, n], {n, 50}]

%t Join[{0},Table[Binomial[2n-2,n-1] (n^2-2n+2)/n,{n,30}]] (* _Harvey P. Dale_, Sep 22 2019 *)

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

%K nonn

%O 0,3

%A _Robert G. Wilson v_, Aug 22 2010

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Last modified May 12 02:23 EDT 2021. Contains 343808 sequences. (Running on oeis4.)