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a(n) = (n^2)*( n^6 + 28*n^4 + 154*n^2 + 132 )/315.
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%I #32 Jun 09 2023 08:53:30

%S 0,1,16,129,704,2945,10128,29953,78592,187137,411280,845185,1640640,

%T 3032705,5373200,9173505,15158272,24331777,38058768,58161793,87037120,

%U 127791489,184402064,261902081,366594816,506298625,690625936,931299201,1242506944,1641303169,2148053520

%N a(n) = (n^2)*( n^6 + 28*n^4 + 154*n^2 + 132 )/315.

%D H. S. M. Coxeter, Regular Polytopes, New York: Dover, 1973.

%H Seiichi Manyama, <a href="/A099195/b099195.txt">Table of n, a(n) for n = 0..10000</a>

%H Milan Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.

%H Milan Janjic and B. Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5

%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2003), 65-75.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F a(n) = (n^2)*( n^6 + 28*n^4 + 154*n^2 + 132 )/315.

%F G.f.: x*(1+x)^7/(1-x)^9. [_R. J. Mathar_, Jul 18 2009]

%F a(n) = 16*a(n-1)/(n-1) + a(n-2) for n > 1. - _Seiichi Manyama_, Jun 06 2018

%t LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,1,16,129,704,2945,10128,29953,78592},40] (* _Harvey P. Dale_, Jan 23 2019 *)

%o (PARI) concat(0, Vec(x*(1+x)^7/(1-x)^9 + O(x^40))) \\ _Michel Marcus_, Dec 14 2015

%Y Similar sequences: A005900 (m=3), A014820(n-1) (m=4), A069038 (m=5), A069039 (m=6), A099193 (m=7), A099196 (m=9), A099197 (m=10).

%Y Cf. A000332.

%K easy,nonn

%O 0,3

%A _Jonathan Vos Post_, Nov 16 2004