A016829 a(n) = (4n+2)^5.
32, 7776, 100000, 537824, 1889568, 5153632, 11881376, 24300000, 45435424, 79235168, 130691232, 205962976, 312500000, 459165024, 656356768, 916132832, 1252332576, 1680700000, 2219006624, 2887174368, 3707398432, 4704270176, 5904900000, 7339040224, 9039207968, 11040808032
Offset: 0
Links
- Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
Programs
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Mathematica
(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 *)
Formula
From Harvey P. Dale, Aug 31 2011: (Start)
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).
G.f.: (32*(x+1)*(x*(x*(x*(x+236)+1446)+236)+1))/(x-1)^6. (End)
From Amiram Eldar, Apr 21 2023: (Start)
a(n) = A016825(n)^5.
a(n) = 2^5*A016757(n).
Sum_{n>=0} 1/a(n) = 31*zeta(5)/1024.
Sum_{n>=0} (-1)^n/a(n) = 5*Pi^5/49152. (End)