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

0,7

Comments

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

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

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

Example

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].

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

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 *)

Prog

(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 */

Crossrefs

Sequence in context: A029152 A320279 A175290 * A162932 A216036 A008924

Adjacent sequences: A089955 A089956 A089957 * A089959 A089960 A089961

Keyword

nonn

Author

Eric W. Weisstein, Nov 16 2003

Status

approved