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%I M4717 #31 Jun 28 2023 20:44:42
%S 0,1,10,57,272,885,2226,4725,8912,15417,24970,38401,56640,80717,
%T 111762,151005,199776,259505,331722,418057,520240,640101,779570,
%U 940677,1125552,1336425,1575626,1845585,2148832,2487997,2865810
%N (25*n^4-120*n^3+209*n^2-108*n)/6.
%D M. Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246, gives this as the number of ways to color faces of a cube using at most n colors, but the formula is incorrect (it was corrected in the second printing) - see A047780.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Harvey P. Dale, <a href="/A006529/b006529.txt">Table of n, a(n) for n = 0..1000</a>
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).
%F a(0)=0, a(1)=1, a(2)=10, a(3)=57, a(4)=272, a(n)=5*a(n-1)- 10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5) [From Harvey P. Dale, Oct 30 2011]
%F G.f.: (-77*x^4-17*x^3-5*x^2-x)/(x-1)^5. - _Harvey P. Dale_, Oct 30 2011
%p A006529:=-z*(1+5*z+17*z**2+77*z**3)/(z-1)**5; [Conjectured by _Simon Plouffe_ in his 1992 dissertation.]
%t Table[(25n^4-120n^3+209n^2-108n)/6,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{0,1,10,57,272},40] (* _Harvey P. Dale_, Oct 30 2011 *)
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_.
%E _Jud McCranie_ noticed this error and gave the correct version of this sequence (A047780).
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