A008674 Expansion of 1/((1-x)*(1-x^3)*(1-x^5)*(1-x^7)*(1-x^9)).

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 11, 14, 16, 19, 23, 26, 30, 35, 40, 45, 52, 58, 65, 74, 82, 91, 102, 113, 124, 138, 151, 165, 182, 198, 216, 236, 256, 277, 301, 325, 350, 379, 407, 437, 471, 504, 539, 578, 617, 658, 703, 748, 795, 847, 899, 953, 1012, 1071, 1133, 1200, 1267, 1337, 1413, 1489, 1568, 1653

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of partitions of n into odd parts <= 9. - Seiichi Manyama, Jun 04 2017

Number of partitions (d1,d2,...,d5) of n such that 0 <= d1/1 <= d2/2 <= ... <= d5/5. - Seiichi Manyama, Jun 04 2017

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 244
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,1,-1,1,-2,2,-2,1,-2,2,-1,2,-2,2,-1,1,-1,1,-1,0,-1,1).

Crossrefs

Cf. A259094.

Programs

Magma

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/&*[1-x^(2*j+1): j in [0..4]] )); // G. C. Greubel, Sep 08 2019

Maple

seq(coeff(series(1/mul(1-x^(2*j+1), j=0..4), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 08 2019

Mathematica

CoefficientList[Series[1/((1-x)(1-x^3)(1-x^5)(1-x^7)(1-x^9)), {x,0,70}], x] (* Vincenzo Librandi, Jun 22 2013 *)
LinearRecurrence[{1,0,1,-1,1,-1,1,-2,2,-2,1,-2,2,-1,2,-2,2,-1,1,-1,1,-1, 0,-1,1}, {1,1,1,2,2,3,4,5,6,8,10,11,14,16,19,23,26,30,35,40,45,52,58, 65,74}, 70] (* Harvey P. Dale, Aug 13 2016 *)

PARI

my(x='x+O('x^70)); Vec(1/prod(j=0,4,1-x^(2*j+1)) ) \\ G. C. Greubel, Sep 08 2019

PARI

a(n) = (n^4+50*n^3+855*n^2+6030*n - n*280*(n%3) +23800)\22680 \\ Hoang Xuan Thanh, Aug 12 2025

Sage

def A008674_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/prod(1-x^(2*j+1) for j in (0..4)) ).list()
A008674_list(70) # G. C. Greubel, Sep 08 2019

Formula

a(n) = floor((n^4+50*n^3+855*n^2+6030*n+23800)/22680 - n*(n mod 3)/81). - Hoang Xuan Thanh, Aug 12 2025

Extensions

Typo in name fixed by Vincenzo Librandi, Jun 22 2013