A008675 Expansion of 1/( Product_{j=0..5} (1-x^(2*j+1)) ).

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 17, 21, 25, 29, 34, 40, 46, 53, 62, 70, 80, 91, 103, 116, 131, 147, 164, 184, 204, 227, 252, 278, 307, 339, 372, 408, 448, 489, 534, 583, 634, 689, 749, 811, 878, 950, 1025, 1106, 1192, 1282, 1378, 1481, 1588, 1702, 1823, 1949, 2083

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of partitions of n into parts 1, 3, 5, 7, 9, and 11. - Joerg Arndt, Jul 09 2013

Number of partitions (d1,d2,...,d6) of n such that 0 <= d1/1 <= d2/2 <= ... <= d6/6. - 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 246
• Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 2, -2, 3, -3, 3, -2, 3, -3, 3, -2, 2, -3, 2, -2, 2, -2, 1, -1, 1, -1, 1, 0, 1, -1).

Crossrefs

Cf. A259094.

Programs

Magma

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( 1/&*[1-x^(2*j+1): j in [0..5]] )); // G. C. Greubel, Sep 08 2019

Maple

seq(coeff(series(1/mul(1-x^(2*j+1), j=0..5), x, n+1), x, n), n = 0..65); # G. C. Greubel, Sep 08 2019

Mathematica

CoefficientList[Series[1/((1-x)(1-x^3)(1-x^5)(1-x^7)(1-x^9)(1-x^11)), {x, 0, 65}], x] (* Vincenzo Librandi, Jun 23 2013 *)
LinearRecurrence[{1,0,1,-1,1,-1,1,-2,2,-2,2,-3,2,-2,3,-3,3,-2,3,-3,3,-2,2,-3,2,-2,2,-2,1,-1,1,-1,1,0,1,-1},{1,1,1,2,2,3,4,5,6,8,10,12,15,17,21,25,29,34,40,46,53,62,70,80,91,103,116,131,147,164,184,204,227,252,278,307},60] (* Harvey P. Dale, Oct 29 2022 *)

PARI

a(n)=(46200*((n\3+1)*[2,-1,-1][n%3+1]+[10,-4,-7][n%3+1]) +3*n^5+ 270*n^4+9005*n^3+136350*n^2+908260*n+3603600)\3742200  \\ Tani Akinari, Jul 09 2013

PARI

Vec(1/((1-x)*(1-x^3)*(1-x^5)*(1-x^7)*(1-x^9)*(1-x^11))+O(x^66)) \\ Joerg Arndt, Jul 09 2013

Sage

def A008674_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/prod(1-x^(2*j+1) for j in (0..5)) ).list()
A008674_list(65) # G. C. Greubel, Sep 08 2019

Extensions

Typo in name fixed by Vincenzo Librandi, Jun 23 2013