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Expansion of g.f.: x/((1-x^2)^5 - 1 + x).
1

%I #17 Sep 08 2022 08:45:28

%S 1,5,25,115,525,2385,10825,49120,222875,1011251,4588335,20818575,

%T 94459755,428590575,1944636420,8823364350,40034094615,181645987625,

%U 824179118751,3739533301365,16967318139775,76985511735170,349304997307275,1584895370489480

%N Expansion of g.f.: x/((1-x^2)^5 - 1 + x).

%H G. C. Greubel, <a href="/A123890/b123890.txt">Table of n, a(n) for n = 0..1000</a>

%H A. Burstein and T. Mansour, <a href="https://arxiv.org/abs/math/0112281">Words restricted by 3-letter ...</a>, arXiv:math/0112281 [math.CO], 2001.

%H A. Burstein and T. Mansour, <a href="https://doi.org/10.1007/s000260300000">Words Restricted by 3-Letter Generalized Multipermutation Patterns</a>, Annals. Combin., 7 (2003), 1-14.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (5,0,-10,0,10,0,-5,0,1).

%p seq(coeff(series(1/(1-5*x+10*x^3-10*x^5+5*x^7-x^9), x, n+1), x, n), n = 0 .. 30); # _G. C. Greubel_, Aug 07 2019

%t CoefficientList[Series[x/((1-x^2)^5 -1+x), {x,0,30}], x] (* _G. C. Greubel_, Aug 07 2019 *)

%o (PARI) my(x='x+O('x^30)); Vec(x/((1-x^2)^5 -1+x)) \\ _G. C. Greubel_, Aug 07 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x/((1-x^2)^5 -1+x) )); // _G. C. Greubel_, Aug 07 2019

%o (Sage)

%o def A123890_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( x/((1-x^2)^5 -1+x) ).list()

%o A123890_list(30) # _G. C. Greubel_, Aug 07 2019

%o (GAP) a:=[1,5,25,115,525,2385,10825,49120,222875];; for n in [10..30] do a[n]:=5*a[n-1]-10*a[n-3] +10*a[n-5]-5*a[n-7]+a[n-9]; od; a; # _G. C. Greubel_, Aug 07 2019

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Nov 20 2006