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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
 0, 1, 2, 10, 50, 238, 1092, 4884, 21450, 92950, 398684, 1696396, 7171892, 30161740, 126293000, 526864680, 2191034970, 9086921190, 37596989100, 155232577500, 639749274780, 2632212288420, 10814090022840, 44369043365400 (list; graph; refs; listen; history; text; internal format)
 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 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: A337348 A218778 A320521 * A015945 A015954 A015949 Adjacent sequences:  A180263 A180264 A180265 * A180267 A180268 A180269 KEYWORD nonn AUTHOR Robert G. Wilson v, Aug 22 2010 STATUS approved

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Last modified April 16 03:25 EDT 2021. Contains 343030 sequences. (Running on oeis4.)