A008762 Expansion of (1+x)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).

1, 2, 3, 5, 8, 11, 15, 20, 26, 33, 41, 50, 61, 73, 86, 101, 118, 136, 156, 178, 202, 228, 256, 286, 319, 354, 391, 431, 474, 519, 567, 618, 672, 729, 789, 852, 919, 989, 1062, 1139, 1220, 1304, 1392, 1484, 1580, 1680, 1784, 1892, 2005, 2122, 2243, 2369, 2500, 2635, 2775, 2920, 3070

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1,-1,1,-1,2,-1).

Programs

GAP

a:=[1,2,3,5,8,11,15,20,26];; for n in [10..60] do a[n]:=2*a[n-1] -a[n-2]+a[n-3]-a[n-4]-a[n-5]+a[n-6]-a[n-7]+2*a[n-8]-a[n-9]; od; a; # G. C. Greubel, Sep 09 2019

Magma

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x)/(&*[1-x^j: j in [1..4]]) )); // G. C. Greubel, Sep 09 2019

Maple

seq(coeff(series( (1+x)/mul(1-x^j, j=1..4) , x, n+1), x, n), n = 0..60); # G. C. Greubel, Sep 09 2019

Mathematica

CoefficientList[Series[(x+1)/Times@@(1-x^Range[4]),{x,0,60}],x] (* or *) LinearRecurrence[{2,-1,1,-1,-1,1,-1,2,-1},{1,2,3,5,8,11,15,20,26},60] (* Harvey P. Dale, Mar 19 2013 *)

PARI

my(x='x+O('x^60)); Vec( (1+x)/prod(j=1,4,1-x^j) ) \\ G. C. Greubel, Sep 09 2019

Sage

def AA008762_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)/prod(1-x^j for j in (1..4)) ).list()
AA008762_list(60) # G. C. Greubel, Sep 09 2019

Formula

a(0)=1, a(1)=2, a(2)=3, a(3)=5, a(4)=8, a(5)=11, a(6)=15, a(7)=20, a(8)=26, a(n) = 2*a(n-1) -a(n-2) +a(n-3) -a(n-4) -a(n-5) +a(n-6) -a(n-7) +2*a(n-8) -a(n-9). - Harvey P. Dale, Mar 19 2013

G.f.: 1/( (1+x)*(1+x^2)*(1+x+x^2)*(1-x)^4 ). - R. J. Mathar, Aug 06 2013

Extensions

Terms a(43) onward added by G. C. Greubel, Sep 09 2019