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%I #56 Sep 08 2022 08:44:36
%S 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,
%T 33,35,37,40,42,44,47,49,52,55,57,60,63,66,69,72,75,78,82,85,88,92,95,
%U 99,103,106,110,114,118,122,126,130,134,139,143,147,152,156,161,166
%N Expansion of 1/((1-x)*(1-x^3)*(1-x^5)).
%C Number of partitions of n into odd parts less than or equal to 5.
%C 1/((1-x^2)*(1-x^6)*(1-x^10)) is the Molien series for the icosahedral group [3,5] of order 120.
%C Number of partitions (d1,d2,d3) of n such that 0 <= d1/1 <= d2/2 <= d3/3. - _Seiichi Manyama_, Jun 04 2017
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, D(n;1,3,5).
%D W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 164 etc.
%D F. Hirzebruch, Letter to _N. J. A. Sloane_, quoted in Ges. Abh. II, 796-798.
%D F. Klein, Lectures on the Icosahedron ..., 2nd Rev. Ed., 1913; reprinted by Dover, NY, 1956; see pp. 236-243.
%D L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 23).
%H Vincenzo Librandi, <a href="/A008672/b008672.txt">Table of n, a(n) for n = 0..10000</a>
%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H G. E. Andrews, P. Paule, A. Riese and V. Strehl, <a href="http://www.risc.uni-linz.ac.at/research/combinat/risc/publications/#ppaule">MacMahon's partition analysis V. Bijections, recursions and magic squares</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=218">Encyclopedia of Combinatorial Structures 218</a>
%H J. S. Leon, V. S. Pless and N. J. A. Sloane, <a href="http://dx.doi.org/10.1016/0097-3165(82)90019-X">Self-dual codes over GF(5)</a>, J. Combin. Theory, A 32 (1982), 178-194.
%H F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, <a href="http://neilsloane.com/doc/gleason2.html">Generalizations of Gleason's theorem on weight enumerators of self-dual codes</a>, IEEE Trans. Inform. Theory, 18 (1972), 794-805; see p. 802, col. 2, foot.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1,1,-1,0,-1,1).
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%F a(n) = round((n+3)*(n+6)/30).
%F a(n) = A025799(2n).
%F a(n) = floor(n^2/30 + 3*n/10 + 1). - _Michael Somos_, Nov 25 2002
%F G.f.: 1/((1-x)*(1-x^3)*(1-x^5)).
%F a(n) = a(-9 - n). - _Michael Somos_, Nov 16 2005
%F 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
%e 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 + ...
%p 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
%t 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 *)
%o (PARI) {a(n) = (n^2 + 9*n)\30 + 1} /* _Michael Somos_, Nov 25 2002 */
%o (Magma) [Round((n+3)*(n+6)/30): n in [0..60]]; // _Vincenzo Librandi_, Jun 23 2011
%o (Sage) [floor((n^2+9*n+30)/30) for n in (0..70)] # _G. C. Greubel_, Sep 08 2019
%o (GAP) List([0..70], n-> Int((n^2+9*n+30)/30) ); # _G. C. Greubel_, Sep 08 2019
%Y Cf. A025799, A259094.
%K nonn,nice,easy
%O 0,4
%A _N. J. A. Sloane_
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