A305724 - OEIS

%I #24 Jun 10 2018 01:43:25

%S 1,201,7001,104881,927441,5707449,26986089,104535009,346615329,

%T 1014889769,2684641785,6526963345,14778775025,31490462745,63670078985,

%U 122977987009,228167048769,408511495049,708522994329,1194315679089,1962053519121,3148993975161

%N Crystal ball sequence for the lattice C_10.

%C Partial sums of A035747.

%H Seiichi Manyama, <a href="/A305724/b305724.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

%F a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n>10.

%F a(n) = Sum_{k=0..10} binomial(20, 2k)*binomial(n+k, 10).

%F G.f.: (1 + 6*x + x^2)*(1 + 184*x + 3740*x^2 + 16136*x^3 + 25414*x^4 + 16136*x^5 + 3740*x^6 + 184*x^7 + x^8) / (1 - x)^11. - _Colin Barker_, Jun 09 2018

%o (PARI) {a(n) = sum(k=0, 10, binomial(20, 2*k)*binomial(n+k, 10))}

%o (PARI) Vec((1 + 6*x + x^2)*(1 + 184*x + 3740*x^2 + 16136*x^3 + 25414*x^4 + 16136*x^5 + 3740*x^6 + 184*x^7 + x^8) / (1 - x)^11 + O(x^30)) \\ _Colin Barker_, Jun 09 2018

%o (GAP) b:=10;; List([0..25],n->Sum([0..b],k->Binomial(2*b,2*k)*Binomial(n+k,b))); # _Muniru A Asiru_, Jun 09 2018

%Y Cf. A035747, A142992.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jun 09 2018