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Expansion of x/((1-x)(1-4x^2)(1-5x)).
(Formerly M4216 N1759)
1

%I M4216 N1759 #46 Aug 27 2022 19:00:51

%S 1,6,35,180,921,4626,23215,116160,581141,2906046,14531595,72659340,

%T 363302161,1816516266,9082603175,45413037720,227065275981,

%U 1135326467286,5676632685955,28383163779300,141915820294601,709579102871106,3547895519947935,17739477605332080,88697388049030021

%N Expansion of x/((1-x)(1-4x^2)(1-5x)).

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A002041/b002041.txt">Table of n, a(n) for n = 1..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 G. B. M. Zerr et al., <a href="https://www.jstor.org/stable/2971110">Problem 64</a>, Amer. Math. Monthly, 3 (1896), 244-248.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1,-24,20).

%F a(n-2) = (1/252) {3*5^n - 4^[(n+2)/2] - 5*4^[(n+3)/2] + 21}. - _Ralf Stephan_, Aug 22 2004

%p A002041:=-1/(z-1)/(2*z+1)/(2*z-1)/(5*z-1); # conjectured by _Simon Plouffe_ in his 1992 dissertation

%t CoefficientList[Series[x/(1-x)/(1-4x^2)/(1-5x),{x,1,30}],x] (* _Vincenzo Librandi_, Jun 12 2012 *)

%t LinearRecurrence[{6,-1,-24,20},{0,1,6,35},30] (* _Harvey P. Dale_, Aug 27 2022 *)

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E Extended by _Vincenzo Librandi_, Jun 12 2012