A060024 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 5.

1, 0, 1, 1, 2, 1, 2, 1, 2, 0, 0, -3, -3, -8, -10, -16, -20, -29, -35, -47, -56, -72, -85, -105, -122, -148, -171, -202, -231, -270, -306, -353, -397, -453, -507, -573, -637, -715, -791, -881, -970, -1075, -1178, -1298, -1417, -1554, -1691, -1846, -2001, -2177, -2353, -2550, -2748, -2969

Offset: 0

Author

N. J. A. Sloane, Mar 17 2001

Keywords

• sign
• easy

Comments

Difference of the number of partitions of n+4 into 4 parts and the number of partitions of n+4 into 5 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,-1,-1,1,1,1,0,0,-1,-1,1).

Crossrefs

Cf. A026810, A026811.

Cf. For other values of N: A060022 (N=3), A060023 (N=4), this sequence (N=5), A060025 (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

Programs

Mathematica

CoefficientList[Series[(1-x-x^5)/(Times@@(1-x^Range[5])),{x,0,60}],x] (* or *) LinearRecurrence[{1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1},{1,0,1,1,2,1,2,1,2,0,0,-3,-3,-8,-10},60] (* Harvey P. Dale, Dec 21 2015 *)

PARI

Vec((1 - x + x^2)*(1 - x^2 - x^3) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)) + O(x^40)) \\ Colin Barker, Apr 17 2019

Formula

a(0)=1, a(1)=0, a(2)=1, a(3)=1, a(4)=2, a(5)=1, a(6)=2, a(7)=1, a(8)=2, a(9)=0, a(10)=0, a(11)=-3, a(12)=-3, a(13)=-8, a(14)=-10, a(n) = a(n-1)+ a(n-2)-a(n-5)-a(n-6)-a(n-7)+a(n-8)+a(n-9)+a(n-10)-a(n-13)- a(n-14)+ a(n-15). - Harvey P. Dale, Dec 21 2015

a(n) = A026810(n+4) - A026811(n+4). - Wesley Ivan Hurt, Apr 16 2019

G.f.: (1 - x + x^2)*(1 - x^2 - x^3) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Apr 17 2019