This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A305721 #46 Mar 15 2024 07:26:08
%S A305721 1,99,1765,14407,74313,284075,880685,2340495,5529233,11905267,
%T A305721 23784309,44673751,79684825,136030779,223619261,355747103,549905697,
%U A305721 828705155,1220925445,1762702695,2498858857,3484382923,4786071885,6484339631,8675201969,11472445971,15009991829
%N A305721 Crystal ball sequence for the lattice C_7.
%C A305721 Partial sums of A019563.
%H A305721 Seiichi Manyama, <a href="/A305721/b305721.txt">Table of n, a(n) for n = 0..10000</a>
%H A305721 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, pp. 727-762.
%H A305721 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F A305721 a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), for n > 7.
%F A305721 a(n) = Sum_{k = 0..7} binomial(14, 2*k)*binomial(n+k, 7).
%F A305721 G.f.: (1 + x)*(1 + 90*x + 911*x^2 + 2092*x^3 + 911*x^4 + 90*x^5 + x^6) / (1 - x)^8. - _Colin Barker_, Jun 09 2018
%F A305721 From _Peter Bala_, Mar 12 2024: (Start)
%F A305721 Sum_{k >= 1} 1/(k*a(k)*a(k-1)) = 2*ln(2) - 289/210 = 1/(99 - 3/(107 - 60/(123 - 315/(147 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*7^2 - ...))))).
%F A305721 E.g.f.: exp(x)*(1 + 98*x + 1568*x^2/2! + 9408*x^3/3! + 26880*x^4/4! + 39424*x^5/5! + 28672*x^6/6! + 8192*x^7/7!).
%F A305721 Note that -T(14, i*sqrt(x)) = 1 + 98*x + 1568*x^2 + 9408*x^3 + 26880*x^4 + 39424*x^5 + 28672*x^6 + 8192*x^7, where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. See A008310.
%F A305721 Row 7 of A142992. (End)
%t A305721 Array[Sum[Binomial[14, 2 k] Binomial[# + k, 7], {k, 0, 7}] &, 27, 0] (* _Michael De Vlieger_, Jun 11 2018 *)
%t A305721 LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,99,1765,14407,74313,284075,880685,2340495},30] (* _Harvey P. Dale_, May 16 2023 *)
%o A305721 (PARI) {a(n) = sum(k=0, 7, binomial(14, 2*k)*binomial(n+k, 7))}
%o A305721 (PARI) Vec((1 + x)*(1 + 90*x + 911*x^2 + 2092*x^3 + 911*x^4 + 90*x^5 + x^6) / (1 - x)^8 + O(x^40)) \\ _Colin Barker_, Jun 09 2018
%o A305721 (GAP) b:=7;; List([0..30],n->Sum([0..b],k->Binomial(2*b,2*k)*Binomial(n+k,b))); # _Muniru A Asiru_, Jun 09 2018
%Y A305721 Cf. A019563, A142992.
%K A305721 nonn,easy
%O A305721 0,2
%A A305721 _Seiichi Manyama_, Jun 09 2018