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a(n) = (4n+2)^5.
1

%I #16 Apr 21 2023 05:11:40

%S 32,7776,100000,537824,1889568,5153632,11881376,24300000,45435424,

%T 79235168,130691232,205962976,312500000,459165024,656356768,916132832,

%U 1252332576,1680700000,2219006624,2887174368,3707398432,4704270176,5904900000,7339040224,9039207968,11040808032

%N a(n) = (4n+2)^5.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F From _Harvey P. Dale_, Aug 31 2011: (Start)

%F a(0)=32, a(1)=7776, a(2)=100000, a(3)=537824, a(4)=1889568, a(5)=5153632, a(n)=6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)- a(n-6).

%F G.f.: (32*(x+1)*(x*(x*(x*(x+236)+1446)+236)+1))/(x-1)^6. (End)

%F From _Amiram Eldar_, Apr 21 2023: (Start)

%F a(n) = A016825(n)^5.

%F a(n) = 2^5*A016757(n).

%F Sum_{n>=0} 1/a(n) = 31*zeta(5)/1024.

%F Sum_{n>=0} (-1)^n/a(n) = 5*Pi^5/49152. (End)

%t (4Range[0,20]+2)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{32,7776,100000,537824,1889568,5153632},20] (* _Harvey P. Dale_, Aug 31 2011 *)

%Y Cf. A013663, A016757, A016825.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_