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%I #14 Sep 08 2022 08:46:04
%S 11,37,87,172,305,501,777,1152,1647,2285,3091,4092,5317,6797,8565,
%T 10656,13107,15957,19247,23020,27321,32197,37697,43872,50775,58461,
%U 66987,76412,86797,98205,110701,124352,139227,155397,172935,191916,212417,234517,258297,283840,311231
%N a(n) = (n+1)*(n^3+15*n^2+74*n+132)/12.
%H Vincenzo Librandi, <a href="/A217947/b217947.txt">Table of n, a(n) for n = 0..1000</a>
%H Ping Sun, <a href="http://dx.doi.org/10.1016/j.disc.2012.09.003">Proof of two conjectures of Petkovsek and Wilf on Gessel walks</a> Discrete Math. 312 (2012), no. 24, 3649--3655. MR2979494. See Th. 1.2, case 3.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: (11-18*x+12*x^2-3*x^3)/(1-x)^5. - _Vincenzo Librandi_, Dec 12 2014
%F a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) n>4. - _Vincenzo Librandi_, Dec 12 2014
%t Table[(n + 1) (n^3 + 15 n^2 + 74 n + 132) / 12, {n, 0, 50}] (* or *) CoefficientList[Series[(11 - 18 x + 12 x^2 - 3 x^3) / (1 - x)^5, {x, 0, 50}], x] (* _Vincenzo Librandi_, Dec 12 2014 *)
%o (Maxima) A217947(n):=(n+1)*(n^3+15*n^2+74*n+132)/12$
%o makelist(A217947(n),n,0,30); /* _Martin Ettl_, Nov 08 2012 */
%o (Magma) [(n+1)*(n^3+15*n^2+74*n+132)/12: n in [0..50]] /* or */ I:=[11,37,87,172,305]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]] ; // _Vincenzo Librandi_, Dec 12 2014
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_, Nov 07 2012