A027339 Number of partitions of n that do not contain 5 as a part.

1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 35, 45, 62, 79, 105, 134, 175, 220, 284, 355, 451, 561, 705, 870, 1085, 1331, 1644, 2008, 2463, 2990, 3646, 4406, 5339, 6425, 7745, 9279, 11135, 13288, 15872, 18875, 22455, 26606, 31537, 37246, 43990, 51796, 60975

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Crossrefs

Column 5 of A175788.

Programs

Mathematica

Table[Count[IntegerPartitions[n],?(FreeQ[#,5]&)],{n,0,50}] (* _Harvey P. Dale, Jul 11 2018 *)

PARI

a(n)=if(n<0,0,polcoeff((1-x^5)/eta(x+x*O(x^n)),n))

Formula

G.f.: (1-x^5) Product_{m>0} 1/(1-x^m).

a(n) = A000041(n)-A000041(n-5).

a(n) ~ 5*Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 5*Pi/(2*sqrt(6)))/sqrt(n) + (61/8 + 9/(2*Pi^2) + 4921*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

Extensions

More terms from Benoit Cloitre, Dec 10 2002