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A305549 Crystal ball sequence for the lattice C_6.

%I A305549 #76 Mar 15 2024 07:26:36
%S A305549 1,73,985,6321,26577,85305,227305,528865,1110049,2149033,3898489,
%T A305549 6704017,11024625,17455257,26751369,39855553,57926209,82368265,
%U A305549 114865945,157417585,212372497,282469881,370879785,481246113,617731681,785065321,988591033,1234319185
%N A305549 Crystal ball sequence for the lattice C_6.
%C A305549 Partial sums of A019562.
%H A305549 Seiichi Manyama, <a href="/A305549/b305549.txt">Table of n, a(n) for n = 0..10000</a>
%H A305549 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 A305549 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A305549 a(n) = (128*n^6 + 384*n^5 + 800*n^4 + 960*n^3 + 692*n^2 + 276*n + 45)/45.
%F A305549 a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7), for n > 6.
%F A305549 a(n) = Sum_{k = 0..6} binomial(12, 2*k)*binomial(n+k, 6).
%F A305549 G.f.: (1 + 6*x + x^2)*(1 + 60*x + 134*x^2 + 60*x^3 + x^4) / (1 - x)^7. - _Colin Barker_, Jun 09 2018
%F A305549 From _Peter Bala_, Mar 12 2024: (Start)
%F A305549 Sum_{k >= 1} 1/(k*a(k)*a(k-1)) = 7/5 - 2*log(2) = 1/(73 - 3/(81 - 60/(97 - 315/(121 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*6^2 - ...))))).
%F A305549 E.g.f.: exp(x)*(1 + 72*x + 840*x^2/2! + 3584*x^3/3! + 6912*x^4/4! + 6144*x^5/5! + 2048*x^6/6!).
%F A305549 Note that T(12, i*sqrt(x)) = 1 + 72*x + 840*x^2 + 3584*x^3 + 6912*x^4 + 6144*x^5 + 2048*x^6, where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. See A008310.
%F A305549 Row 6 of A142992. (End)
%o A305549 (PARI) {a(n) = sum(k=0, 6, binomial(12,2*k)*binomial(n+k,6))}
%o A305549 (PARI) Vec((1 + 6*x + x^2)*(1 + 60*x + 134*x^2 + 60*x^3 + x^4) / (1 - x)^7 + O(x^40)) \\ _Colin Barker_, Jun 09 2018
%Y A305549 Cf. A019562, A142992.
%K A305549 nonn,easy
%O A305549 0,2
%A A305549 _Seiichi Manyama_, Jun 09 2018