A008672 Expansion of 1/((1-x)*(1-x^3)*(1-x^5)).

1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 20, 22, 23, 25, 27, 29, 31, 33, 35, 37, 40, 42, 44, 47, 49, 52, 55, 57, 60, 63, 66, 69, 72, 75, 78, 82, 85, 88, 92, 95, 99, 103, 106, 110, 114, 118, 122, 126, 130, 134, 139, 143, 147, 152, 156, 161, 166

Offset

0,4

Comments

Number of partitions of n into odd parts less than or equal to 5.

1/((1-x^2)*(1-x^6)*(1-x^10)) is the Molien series for the icosahedral group [3,5] of order 120.

Number of partitions (d1,d2,d3) of n such that 0 <= d1/1 <= d2/2 <= d3/3. - Seiichi Manyama, Jun 04 2017

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, D(n;1,3,5).

W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 164 etc.

F. Hirzebruch, Letter to N. J. A. Sloane, quoted in Ges. Abh. II, 796-798.

F. Klein, Lectures on the Icosahedron ..., 2nd Rev. Ed., 1913; reprinted by Dover, NY, 1956; see pp. 236-243.

L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 23).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

G. E. Andrews, P. Paule, A. Riese and V. Strehl, MacMahon's partition analysis V. Bijections, recursions and magic squares

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 218

J. S. Leon, V. S. Pless and N. J. A. Sloane, Self-dual codes over GF(5), J. Combin. Theory, A 32 (1982), 178-194.

F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, Generalizations of Gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, 18 (1972), 794-805; see p. 802, col. 2, foot.

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

Index entries for Molien series

Formula

a(n) = round((n+3)*(n+6)/30).

a(n) = A025799(2n).

a(n) = floor(n^2/30 + 3*n/10 + 1). - Michael Somos, Nov 25 2002

G.f.: 1/((1-x)*(1-x^3)*(1-x^5)).

a(n) = a(-9 - n). - Michael Somos, Nov 16 2005

a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-5) - a(n-6) - a(n-8) + a(n-9); a(0)=1, a(1)=1, a(2)=1, a(3)=2, a(4)=2, a(5)=3, a(6)=4, a(7)=4, a(8)=5. - Harvey P. Dale, Feb 07 2012

Example

G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 6*x^9 + 7*x^10 + ...

Maple

seq(coeff(series(1/((1-x)*(1-x^3)*(1-x^5)), 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)), {x, 0, 70}], x] (* or *) LinearRecurrence[{1, 0, 1, -1, 1, -1, 0, -1, 1}, {1, 1, 1, 2, 2, 3, 4, 4, 5}, 70] (* Harvey P. Dale, Feb 07 2012 *)

Prog

(PARI) {a(n) = (n^2 + 9*n)\30 + 1} /* Michael Somos, Nov 25 2002 */
(Magma) [Round((n+3)*(n+6)/30): n in [0..60]]; // Vincenzo Librandi, Jun 23 2011
(Sage) [floor((n^2+9*n+30)/30) for n in (0..70)] # G. C. Greubel, Sep 08 2019
(GAP) List([0..70], n-> Int((n^2+9*n+30)/30) ); # G. C. Greubel, Sep 08 2019

Crossrefs

Cf. A025799, A259094.

Sequence in context: A097950 A011885 A211524 * A097923 A027582 A259198

Adjacent sequences: A008669 A008670 A008671 * A008673 A008674 A008675

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved