A008766 Expansion of (1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).

1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 29, 36, 45, 54, 65, 77, 91, 106, 123, 141, 162, 184, 208, 234, 263, 293, 326, 361, 399, 439, 482, 527, 576, 627, 681, 738, 799, 862, 929, 999, 1073, 1150, 1231, 1315, 1404, 1496, 1592, 1692, 1797, 1905, 2018, 2135, 2257, 2383, 2514, 2649, 2790, 2935

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From Washington Bomfim, Jan 14 2021: (Start)

Let \n,m\ be the number of partitions of n into m non-distinct parts.

For n >= 1, \n,5\ = round((2*n^3-15*n^2+60*n-110*[n mod 2 = 0]-65*[n mod 2])/144).

For n >= 10, \n,5\ = A026811(n) - A026811(n-10).

(End)

Links

• Washington Bomfim, Table of n, a(n) for n = 0..9999 (first 1000 terms from G. C. Greubel)
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1,-1,1,-1,2,-1).

Crossrefs

Cf. A026811, A001401.

Programs

GAP

a:=[1,1,2,3,5,7,10,13,18];; 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 10 2019

Magma

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) )); // G. C. Greubel, Sep 10 2019

Maple

seq(coeff(series((1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)), x, n+1), x, n), n = 0 .. 60); # G. C. Greubel, Sep 10 2019

Mathematica

CoefficientList[Series[(1+x^5)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4),{x,0,60}],x] (* or *) LinearRecurrence[{2,-1,1,-1,-1,1,-1,2,-1}, {1,1,2,3,5,7,10, 13,18}, 60] (* Harvey P. Dale, Jul 24 2016 *)

PARI

Vec((1+x^5)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4) +O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

PARI

seq(x) = { a = vector(x+1); my(N = 5);
for(n=0,x, a[n+1]=round((2*N^3-15*N^2+60*N-110*!(N%2)-65*(N%2))/144); N++);a};
seq(60) \\ Washington Bomfim, Jan 14 2021

Sage

def A008766_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))).list()
A008766_list(60) # G. C. Greubel, Sep 10 2019

Formula

a(n) = round((2*N^3 - 15*N^2 + 60*N - 110*[N mod 2=0] - 65*[N mod 2])/144), where N = n+5. - Washington Bomfim, Jan 14 2021

Extensions

Terms a(45) onward added by G. C. Greubel, Sep 10 2019