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%I #27 Sep 08 2022 08:45:27
%S 1,5,65,1205,26465,628805,15424865,382964405,9550195265,238539648005,
%T 5961554097665,149021418519605,3725378557692065,93133051794619205,
%U 2328313585536338465,58207725254446186805,1455192101905494196865
%N Number of closed walks of length 2*n on the 5-cube.
%H G. C. Greubel, <a href="/A121822/b121822.txt">Table of n, a(n) for n = 0..700</a>
%H G. R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a number pyramid related to the binomial, Deleham, Eulerian, MacMahon and Stirling number triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
%H L. Reyzin, Mathoverflow, <a href="https://mathoverflow.net/questions/71736/number-of-closed-walks-on-an-n-cube">Number of closed walks on an n-cube</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-259,225).
%F a(n) = (25^n + 5*9^n + 10)/16.
%F G.f.: (1 - 30*x + 149*x^2)/(1 - 35*x + 259*x^2 - 225*x^3).
%F From _Peter Bala_, Nov 13 2006: (Start)
%F E.g.f.: cosh^5(x).
%F O.g.f.: 1/(1-5*1x/(1-4*2x/(1-3*3x/(1-2*4x/(1-1*5x))))) (continued fraction). (End)
%F (-1)^n*a(n) = Sum_{k=0..n} A086872(n,k)*(-6)^(n-k). - _Philippe Deléham_, Aug 17 2007
%F a(n) = (1/2^5)*Sum_{j = 0..5} binomial(5,j)*(5 - 2*j)^(2*n). See Reyzin link. - _Peter Bala_, Jun 03 2019
%t Table[(25^n +5*9^n +10)/16, {n,0,20}] (* _G. C. Greubel_, Jun 07 2019 *)
%o (PARI) a(n)=(25^n+5*9^n+10)>>4 \\ _Charles R Greathouse IV_, Jan 17 2012
%o (Magma) [(25^n +5*9^n +10)/16: n in [0..20]]; // _G. C. Greubel_, Jun 07 2019
%o (Sage) [(25^n +5*9^n +10)/16 for n in (0..20)] # _G. C. Greubel_, Jun 07 2019
%o (GAP) List([0..20], n-> (25^n +5*9^n +10)/16) # _G. C. Greubel_, Jun 07 2019
%Y Cf. A054879, A081294, A092812.
%K nonn,easy
%O 0,2
%A _Philippe Deléham_, Aug 27 2006
%E Corrected by _T. D. Noe_, Nov 07 2006
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