cp's OEIS Frontend

A305723 Crystal ball sequence for the lattice C_9.

%I A305723 #24 Jun 09 2023 16:27:12
%S A305723 1,163,4645,57799,432073,2286955,9446125,32398735,96220561,254831667,
%T A305723 614859189,1373356887,2874747225,5693596923,10751213181,19475555103,
%U A305723 34015593249,57523019715,94516111685,151342583015,236760421097,362658000011,544937185805,804585705647
%N A305723 Crystal ball sequence for the lattice C_9.
%C A305723 Partial sums of A035746.
%H A305723 Seiichi Manyama, <a href="/A305723/b305723.txt">Table of n, a(n) for n = 0..10000</a>
%H A305723 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 A305723 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F A305723 a(n) = Sum_{k=0..9} binomial(18, 2k)*binomial(n+k, 9).
%F A305723 From _Colin Barker_, Jun 09 2018: (Start)
%F A305723 G.f.: (1 + x)*(1 + 14*x + x^2)*(1 + 138*x + 975*x^2 + 1868*x^3 + 975*x^4 + 138*x^5 + x^6) / (1 - x)^10.
%F A305723 a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n>9.
%F A305723 (End)
%t A305723 Table[Sum[Binomial[18,2k]Binomial[n+k,9],{k,0,9}],{n,0,40}] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,163,4645,57799,432073,2286955,9446125,32398735,96220561,254831667},40] (* _Harvey P. Dale_, Jun 09 2023 *)
%o A305723 (PARI) {a(n) = sum(k=0, 9, binomial(18, 2*k)*binomial(n+k, 9))}
%o A305723 (PARI) Vec((1 + x)*(1 + 14*x + x^2)*(1 + 138*x + 975*x^2 + 1868*x^3 + 975*x^4 + 138*x^5 + x^6) / (1 - x)^10 + O(x^40)) \\ _Colin Barker_, Jun 09 2018
%o A305723 (GAP) b:=9;; List([0..25],n->Sum([0..b],k->Binomial(2*b,2*k)*Binomial(n+k,b))); # _Muniru A Asiru_, Jun 09 2018
%Y A305723 Cf. A035746, A142992.
%K A305723 nonn
%O A305723 0,2
%A A305723 _Seiichi Manyama_, Jun 09 2018