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).
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
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved