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A305722 Crystal ball sequence for the lattice C_8.

%I A305722 #29 May 10 2020 08:29:04
%S A305722 1,129,2945,29953,187137,845185,3032705,9173505,24331777,58161793,
%T A305722 127791489,261902081,506298625,931299201,1641303169,2786931713,
%U A305722 4580166657,7312946305,11379709825,17304414465,25772582657,37668968833,54121468545,76551925249
%N A305722 Crystal ball sequence for the lattice C_8.
%C A305722 Partial sums of A019564.
%H A305722 Seiichi Manyama, <a href="/A305722/b305722.txt">Table of n, a(n) for n = 0..10000</a>
%H A305722 R. Bacher, P. de la Harpe and B. Venkov, <a href="https://doi.org/10.5802/aif.1689">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, Annales de l'institut Fourier, Tome 49 (1999) no. 3 , p. 727-762.
%H A305722 <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 A305722 a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9), for n>8.
%F A305722 a(n) = Sum_{k=0..8} binomial(16, 2k)*binomial(n+k, 8).
%F A305722 G.f.: (1 + 120*x + 1820*x^2 + 8008*x^3 + 12870*x^4 + 8008*x^5 + 1820*x^6 + 120*x^7 + x^8) / (1 - x)^9. - _Colin Barker_, Jun 09 2018
%o A305722 (PARI) {a(n) = sum(k=0, 8, binomial(16, 2*k)*binomial(n+k, 8))}
%o A305722 (PARI) Vec((1 + 120*x + 1820*x^2 + 8008*x^3 + 12870*x^4 + 8008*x^5 + 1820*x^6 + 120*x^7 + x^8) / (1 - x)^9 + O(x^40)) \\ _Colin Barker_, Jun 09 2018
%o A305722 (GAP) b:=8;; List([0..25],n->Sum([0..b],k->Binomial(2*b,2*k)*Binomial(n+k,b))); # _Muniru A Asiru_, Jun 09 2018
%Y A305722 Cf. A019564, A142992.
%K A305722 nonn
%O A305722 0,2
%A A305722 _Seiichi Manyama_, Jun 09 2018