A060023 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 4.

1, 0, 1, 1, 1, 0, 1, -1, -1, -3, -4, -7, -8, -13, -15, -20, -24, -31, -35, -44, -50, -60, -68, -80, -89, -104, -115, -131, -145, -164, -179, -201, -219, -243, -264, -291, -314, -345, -371, -404, -434, -471, -503, -544, -580, -624, -664, -712, -755, -808, -855, -911, -963, -1024, -1079, -1145

Offset: 0

Author

N. J. A. Sloane, Mar 17 2001

Keywords

• sign
• easy

Comments

Difference of the number of partitions of n+3 into 3 parts and the number of partitions of n+3 into 4 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, -2, 0, 0, 1, 1, -1).

Crossrefs

Cf. A026810, A069905.

Cf. For other values of N: A060022 (N=3), this sequence (N=4), A060024 (N=5), A060025 (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

Programs

Magma

I:=[1,0,1,1,1,0,1,-1,-1,-3]; [n le 10 select I[n] else Self(n-1)+Self(n-2)-2*Self(n-5)+Self(n-8)+Self(n-9)-Self(n-10): n in [1..60]]; // Vincenzo Librandi, Jun 23 2015

Mathematica

CoefficientList[Series[(1-x-x^4)/Times@@(1-x^Range[4]),{x,0,60}],x] (* or *) LinearRecurrence[{1,1,0,0,-2,0,0,1,1,-1},{1,0,1,1,1,0,1,-1,-1,-3},70] (* Harvey P. Dale, Jan 14 2015 *)

PARI

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

Formula

a(n) = A069905(n+3) - A026810(n+3). - Wesley Ivan Hurt, Apr 16 2019

From Colin Barker, Apr 17 2019: (Start)

G.f.: (1 - x - x^4) / ((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).

a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n>9.

(End)