A008673 Expansion of 1/((1-x)*(1-x^3)*(1-x^5)*(1-x^7)).

1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 16, 19, 21, 24, 27, 30, 34, 38, 42, 46, 51, 56, 61, 67, 73, 79, 86, 93, 100, 108, 116, 125, 134, 143, 153, 163, 174, 185, 197, 209, 221, 235, 248, 262, 277, 292, 308, 324, 341, 358, 376, 395, 414, 434, 454, 475, 497, 519, 542, 566, 590

Offset

0,4

Comments

Number of partitions of n into parts 1, 3, 5, and 7. - Joerg Arndt, Jul 08 2013

Number of partitions (d1,d2,d3,d4) of n such that 0 <= d1/1 <= d2/2 <= d3/3 <= d4/4. - 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 234

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,1,-1,1,-2,1,-1,1,-1,1,0,1,-1).

Formula

a(n) = floor((n^3 + 24*n^2 + 171*n + 630)/630). - Tani Akinari, Jul 08 2013

a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-5) - a(n-6) + a(n-7) - 2*a(n-8) + a(n-9) - a(n-10) + a(n-11) - a(n-12) + a(n-13) + a(n-15) - a(n-16). - David Neil McGrath, Feb 14 2015

Example

There are a(7)=5 partitions of n=7 into parts 1, 3, 5, and 7: (7), (511), (331), (31111), and (1111111). - David Neil McGrath, Feb 14 2015

Maple

seq(coeff(series(1/((1-x)*(1-x^3)*(1-x^5)*(1-x^7)), 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)), {x, 0, 70}], x] (* Vincenzo Librandi, Jun 22 2013 *)
LinearRecurrence[{1, 0, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, 0, 1, -1}, {1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 16, 19}, 70] (* Harvey P. Dale, Jul 08 2019 *)

Prog

(PARI) vector(70, n, m=n-1; (m^3+24*m^2+171*m+630)\630 ) \\ G. C. Greubel, Sep 08 2019
(Magma) [Floor((n^3+24*n^2+171*n+630)/630): n in [0..70]]; // G. C. Greubel, Sep 08 2019
(Sage) [floor((n^3+24*n^2+171*n+630)/630) for n in (0..70)] # G. C. Greubel, Sep 08 2019
(GAP) List([0..70], n-> Int((n^3+24*n^2+171*n+630)/630) ); # G. C. Greubel, Sep 08 2019

Crossrefs

Cf. A259094.

Sequence in context: A065459 A011873 A173151 * A133564 A342558 A017863

Adjacent sequences: A008670 A008671 A008672 * A008674 A008675 A008676

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved