A060025 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 6.

1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 2, 3, -1, -1, -6, -9, -17, -22, -35, -43, -61, -76, -100, -121, -155, -185, -229, -271, -328, -383, -458, -529, -622, -715, -830, -946, -1090, -1233, -1407, -1584, -1794, -2008, -2261, -2517, -2816, -3124, -3476, -3838, -4253, -4677, -5159, -5656, -6213

Offset: 0

Author

N. J. A. Sloane, Mar 17 2001

Keywords

• sign
• easy

Comments

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

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
• Index entries for sequences related to partitions
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1).

Crossrefs

Cf. A026811, A026812.

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).

Programs

Magma

m:=6; R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x-x^m)/( (&*[1-x^j: j in [1..m]]) ) )); // G. C. Greubel, Apr 17 2019

Mathematica

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

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

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

Formula

a(n) = A026811(n+5) - A026812(n+5). - Wesley Ivan Hurt, Apr 16 2019

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