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A008766 Expansion of (1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).

%I A008766 #31 Nov 03 2023 15:10:07
%S A008766 1,1,2,3,5,7,10,13,18,23,29,36,45,54,65,77,91,106,123,141,162,184,208,
%T A008766 234,263,293,326,361,399,439,482,527,576,627,681,738,799,862,929,999,
%U A008766 1073,1150,1231,1315,1404,1496,1592,1692,1797,1905,2018,2135,2257,2383,2514,2649,2790,2935
%N A008766 Expansion of (1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
%C A008766 From _Washington Bomfim_, Jan 14 2021: (Start)
%C A008766 Let \n,m\ be the number of partitions of n into m non-distinct parts.
%C A008766 For n >= 1, \n,5\ = round((2*n^3-15*n^2+60*n-110*[n mod 2 = 0]-65*[n mod 2])/144).
%C A008766 For n >= 10, \n,5\ = A026811(n) - A026811(n-10).
%C A008766 (End)
%H A008766 Washington Bomfim, <a href="/A008766/b008766.txt">Table of n, a(n) for n = 0..9999</a> (first 1000 terms from G. C. Greubel)
%H A008766 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-1,-1,1,-1,2,-1).
%F A008766 a(n) = round((2*N^3 - 15*N^2 + 60*N - 110*[N mod 2=0] - 65*[N mod 2])/144), where N = n+5. - _Washington Bomfim_, Jan 14 2021
%p A008766 seq(coeff(series((1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)), x, n+1), x, n), n = 0 .. 60); # _G. C. Greubel_, Sep 10 2019
%t A008766 CoefficientList[Series[(1+x^5)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4),{x,0,60}],x] (* or *) LinearRecurrence[{2,-1,1,-1,-1,1,-1,2,-1}, {1,1,2,3,5,7,10, 13,18}, 60] (* _Harvey P. Dale_, Jul 24 2016 *)
%o A008766 (PARI) Vec((1+x^5)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4) +O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%o A008766 (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) )); // _G. C. Greubel_, Sep 10 2019
%o A008766 (Sage)
%o A008766 def A008766_list(prec):
%o A008766     P.<x> = PowerSeriesRing(ZZ, prec)
%o A008766     return P((1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))).list()
%o A008766 A008766_list(60) # _G. C. Greubel_, Sep 10 2019
%o A008766 (GAP) a:=[1,1,2,3,5,7,10,13,18];; for n in [10..60] do a[n]:=2*a[n-1]-a[n-2]+a[n-3]-a[n-4]-a[n-5]+a[n-6]-a[n-7]+2*a[n-8]-a[n-9]; od; a; # _G. C. Greubel_, Sep 10 2019
%o A008766 (PARI) seq(x) = { a = vector(x+1); my(N = 5);
%o A008766 for(n=0,x, a[n+1]=round((2*N^3-15*N^2+60*N-110*!(N%2)-65*(N%2))/144); N++);a};
%o A008766 seq(60) \\ _Washington Bomfim_, Jan 14 2021
%Y A008766 Cf. A026811, A001401.
%K A008766 nonn,easy
%O A008766 0,3
%A A008766 _N. J. A. Sloane_
%E A008766 Terms a(45) onward added by _G. C. Greubel_, Sep 10 2019