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%I #31 Sep 08 2022 08:44:41
%S 1,729,15625,117649,531441,1771561,4826809,11390625,24137569,47045881,
%T 85766121,148035889,244140625,387420489,594823321,887503681,
%U 1291467969,1838265625,2565726409,3518743761,4750104241,6321363049,8303765625,10779215329,13841287201,17596287801
%N a(n) = (2*n+1)^6.
%H Vincenzo Librandi, <a href="/A016758/b016758.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(0)=1, a(1)=729, a(2)=15625, a(3)=117649, a(4)=531441, a(5)=1771561, a(6)=4826809, 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). - _Harvey P. Dale_, Dec 26 2012
%F G.f.: (1 +722*x +10543*x^2 +23548*x^3 +10543*x^4 +722*x^5 +x^6)/(1-x)^7 . - _R. J. Mathar_, Jul 07 2017
%F Sum_{n>=0} 1/a(n) = Pi^6/960 (A300709). - _Amiram Eldar_, Oct 10 2020
%F From _Amiram Eldar_, Jan 28 2021: (Start)
%F Product_{n>=0} (1 + 1/a(n)) = cosh(Pi/2)*(cos(sqrt(3)*Pi/2) + cosh(Pi/2))/2.
%F Product_{n>=1} (1 - 1/a(n)) = Pi*cosh(sqrt(3)*Pi/2)/24. (End)
%t (2*Range[0,20]+1)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{1, 729,15625,117649,531441,1771561,4826809},20] (* _Harvey P. Dale_, Dec 26 2012 *)
%o (Magma) [(2*n+1)^6: n in [0..30]]; // _Vincenzo Librandi_, Sep 07 2011
%o (PARI) vector(30, n, n--; (2*n+1)^6) \\ _G. C. Greubel_, Sep 15 2018
%Y Cf. A300709.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_