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Number of partitions of n in which every part occurs 2, 3, or 5 times.
2

%I #32 Apr 21 2016 02:19:04

%S 1,0,1,1,1,1,3,1,3,4,4,4,8,5,9,11,11,12,20,15,23,27,28,31,45,38,52,61,

%T 64,71,96,87,112,129,136,151,194,184,227,259,275,304,376,368,441,499,

%U 531,586,704,705,826,927,989,1088,1280,1302,1500,1672,1787,1960,2267

%N Number of partitions of n in which every part occurs 2, 3, or 5 times.

%C Also number of partitions of n in which every part is congruent to {2, 3, 6, 9, 10} mod 12. - _Vladeta Jovovic_, Jan 07 2005

%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.5).

%D M. V. Subbarao, Combinatorial proofs of some identities, Proc. Washington State Univ. Conf. Number Theory, 1971, pp. 80-91.

%H Alois P. Heinz, <a href="/A089958/b089958.txt">Table of n, a(n) for n = 0..1000</a>

%H M. V. Subbarao, <a href="http://dx.doi.org/10.1090/S0002-9939-1971-0272736-9">On a partition theorem of MacMahon-Andrews</a>, Proc. Amer. Math. Soc., 27, 1971, 449-450.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>

%F Euler transform of period 12 sequence [0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, ...]. - _Vladeta Jovovic_, Jan 07 2005

%F Expansion of q^(-5/24)eta(q^6)eta(q^4)/(eta(q^2)eta(q^3)) in powers of q.

%F G.f.: Product_{j>=1}(1+x^(2j)+x^(3j)+x^(5j)). - _Emeric Deutsch_, Mar 05 2006

%F a(n) ~ 5^(1/4) * exp(Pi*sqrt(5*n/2)/3) / (2^(11/4) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Aug 24 2015

%e a(11) = 4 because we have [4,4,1,1,1], [3,3,3,1,1], [3,3,1,1,1,1,1] and [2,2,2,1,1,1,1,1].

%p g:=product(1+x^(2*j)+x^(3*j)+x^(5*j),j=1..50): gser:=series(g,x=0,63): seq(coeff(gser,x,n),n=0..60); # _Emeric Deutsch_, Mar 05 2006

%t nn = 60; CoefficientList[ Series[Product[1 + x^(2 i) + x^(3 i) + x^(5 i), {i, 1, nn}], {x, 0, nn}], x] (* _Geoffrey Critzer_, May 31 2013 *)

%t QP = QPochhammer; s = QP[q^6]*(QP[q^4]/(QP[q^2]*QP[q^3])) + O[q]^70; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 30 2015, adapted from PARI *)

%o (PARI) {a(n)=local(A); if(n<0,0, A=x*O(x^n); polcoeff( eta(x^4+A) *eta(x^6+A)/eta(x^2+A)/eta(x^3+A), n))} /* _Michael Somos_, Jan 19 2005 */

%K nonn

%O 0,7

%A _Eric W. Weisstein_, Nov 16 2003