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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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

Examples

			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 + ...
		

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

Crossrefs

Cf. A025799, A259094.

Programs

GAP

List([0..70], n-> Int((n^2+9*n+30)/30) ); # G. C. Greubel, Sep 08 2019

Magma

[Round((n+3)*(n+6)/30): n in [0..60]]; // Vincenzo Librandi, Jun 23 2011

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 *)

PARI

{a(n) = (n^2 + 9*n)\30 + 1} /* Michael Somos, Nov 25 2002 */

Sage

[floor((n^2+9*n+30)/30) for n in (0..70)] # G. C. Greubel, Sep 08 2019

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