A029032 Expansion of 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)).

1, 1, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 17, 20, 23, 27, 31, 35, 40, 45, 51, 57, 63, 70, 78, 86, 94, 103, 113, 123, 134, 145, 157, 170, 183, 197, 212, 227, 243, 260, 278, 296, 315, 335, 356, 378, 400, 423, 448, 473, 499, 526, 554, 583, 613, 644, 676, 709, 743

Offset

0,4

Comments

a(n) is the number of partitions of n into parts 1, 3, 4, and 5. - David Neil McGrath, Sep 13 2014

Links

Hoang Xuan Thanh, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,0,-1,-1,0,0,1,0,1,-1).

Formula

a(0)=1, a(1)=1, a(2)=1, a(3)=2, a(4)=3, a(5)=4, a(6)=5, a(7)=6, a(8)=8, a(9)=10, a(10)=12, a(11)=14, a(12)=17, a(n)=a(n-1)+a(n-3)-a(n-6)- a(n-7)+ a(n-10)+a(n-12)-a (n-13). - Harvey P. Dale, Jan 04 2012

From R. J. Mathar, Jun 23 2021: (Start)

a(n)-a(n-1) = A008680(n).

a(n)-a(n-3) = A025772(n).

a(n)-a(n-4) = A008672(n).

a(n)-a(n-5) = A025767(n). (End)

a(n) = 1 + floor((2*n^3+39*n^2+228*n)/720). - Hoang Xuan Thanh, May 29 2025

Maple

M := Matrix(13, (i, j)-> if (i=j-1) or (j=1 and member(i, [1, 3, 10, 12])) then 1 elif j=1 and member(i, [6, 7, 13]) then -1 else 0 fi); a := n -> (M^(n))[1, 1]; seq (a(n), n=0..49); # Alois P. Heinz, Jul 25 2008

Mathematica

CoefficientList[Series[1/((1-x)(1-x^3)(1-x^4)(1-x^5)), {x, 0, 50}], x] (* Harvey P. Dale, Jan 04 2012 *)

Crossrefs

Sequence in context: A237118 A112402 A056864 * A218906 A059809 A327634

Adjacent sequences: A029029 A029030 A029031 * A029033 A029034 A029035

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved