A089958 Number of partitions of n in which every part occurs 2, 3, or 5 times.
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, 64, 71, 96, 87, 112, 129, 136, 151, 194, 184, 227, 259, 275, 304, 376, 368, 441, 499, 531, 586, 704, 705, 826, 927, 989, 1088, 1280, 1302, 1500, 1672, 1787, 1960, 2267
Offset: 0
Keywords
Examples
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].
References
- I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.5).
- M. V. Subbarao, Combinatorial proofs of some identities, Proc. Washington State Univ. Conf. Number Theory, 1971, pp. 80-91.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- M. V. Subbarao, On a partition theorem of MacMahon-Andrews, Proc. Amer. Math. Soc., 27, 1971, 449-450.
- Eric Weisstein's World of Mathematics, Partition Function P
Programs
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Maple
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
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Mathematica
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 *) 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 *) -
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 */
Formula
Euler transform of period 12 sequence [0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, ...]. - Vladeta Jovovic, Jan 07 2005
Expansion of q^(-5/24)eta(q^6)eta(q^4)/(eta(q^2)eta(q^3)) in powers of q.
G.f.: Product_{j>=1}(1+x^(2j)+x^(3j)+x^(5j)). - Emeric Deutsch, Mar 05 2006
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
Comments