A060025 - OEIS

%I #23 Sep 08 2022 08:45:03

%S 1,0,1,1,2,2,3,2,4,3,4,2,3,-1,-1,-6,-9,-17,-22,-35,-43,-61,-76,-100,

%T -121,-155,-185,-229,-271,-328,-383,-458,-529,-622,-715,-830,-946,

%U -1090,-1233,-1407,-1584,-1794,-2008,-2261,-2517,-2816,-3124,-3476,-3838,-4253,-4677,-5159,-5656,-6213

%N Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 6.

%C Difference of the number of partitions of n+5 into 5 parts and the number of partitions of n+5 into 6 parts. - _Wesley Ivan Hurt_, Apr 16 2019

%H Colin Barker, <a href="/A060025/b060025.txt">Table of n, a(n) for n = 0..1000</a>

%H P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s1-17.1.139">Perpetual reciprocants</a>, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1).

%F a(n) = A026811(n+5) - A026812(n+5). - _Wesley Ivan Hurt_, Apr 16 2019

%F G.f.: (1 - x - x^6) / ((1 - x)^6*(1 + x)^3*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)). - _Colin Barker_, Apr 17 2019

%t With[{nn=6},CoefficientList[Series[(1-x-x^nn)/Times@@(1-x^Range[nn]),{x,0,60}],x]] (* _Harvey P. Dale_, May 15 2016 *)

%o (PARI) Vec((1 - x - x^6) / ((1 - x)^6*(1 + x)^3*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ _Colin Barker_, Apr 17 2019

%o (Magma) m:=6; R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x-x^m)/( (&*[1-x^j: j in [1..m]]) ) )); // _G. C. Greubel_, Apr 17 2019

%o (Sage) m=6; ((1-x-x^m)/( product(1-x^j for j in (1..m)) )).series(x, 60).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 17 2019

%Y Cf. A026811, A026812.

%Y Cf. For other values of N: A060022 (N=3), A060023 (N=4), A060024 (N=5), this sequence (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

%K sign,easy

%O 0,5

%A _N. J. A. Sloane_, Mar 17 2001