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A049016 Expansion of 1/((1-x)^5 - x^5).

%I A049016 #30 Apr 11 2023 08:42:18
%S A049016 1,5,15,35,70,127,220,385,715,1430,3004,6385,13380,27370,54740,107883,
%T A049016 211585,416405,826045,1652090,3321891,6690150,13455325,26985675,
%U A049016 53971350,107746282,214978335,429124630,857417220,1714834440,3431847189
%N A049016 Expansion of 1/((1-x)^5 - x^5).
%H A049016 Seiichi Manyama, <a href="/A049016/b049016.txt">Table of n, a(n) for n = 0..3000</a>
%H A049016 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,2).
%F A049016 G.f.: 1/((1-x)^5-x^5) = 1/( (1-2*x)*(1-3*x+4*x^2-2*x^3+x^4) ).
%F A049016 a(10*n+3) = A078789(5*n+3).
%F A049016 a(10*n+5) = A078789(5*n+4).
%F A049016 a(n) = (-1)^n * A000750(n).
%F A049016 Binomial transform of expansion of (1+x)^4/(1-x^5), or (1, 4, 6, 4, 1, 1, 4, 6, 4, 1, ...). - _Paul Barry_, Mar 19 2004
%F A049016 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 2*a(n-5). - _Paul Curtz_, May 24 2008
%F A049016 G.f.: -1/( x^5 - 1 + 5*x/Q(0) ) where Q(k) = 1 + k*(x+1) + 5*x - x*(k+1)*(k+6)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Mar 15 2013
%t A049016 CoefficientList[Series[1/((1-x)^5-x^5),{x,0,30}],x] (* or *) LinearRecurrence[ {5,-10,10,-5,2},{1,5,15,35,70},40] (* _Harvey P. Dale_, Jan 20 2014 *)
%o A049016 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/((1-x)^5-x^5) )); // _G. C. Greubel_, Apr 11 2023
%o A049016 (SageMath)
%o A049016 def A049016_list(prec):
%o A049016     P.<x> = PowerSeriesRing(ZZ, prec)
%o A049016     return P( 1/((1-x)^5-x^5) ).list()
%o A049016 A049016_list(30) # _G. C. Greubel_, Apr 11 2023
%Y A049016 Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), A000749 (m=4), this sequence (m=5), A192080 (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).
%Y A049016 Cf. A000750, A078789.
%K A049016 nonn,easy
%O A049016 0,2
%A A049016 _N. J. A. Sloane_