A027337 Number of partitions of n that do not contain 3 as a part.

1, 1, 2, 2, 4, 5, 8, 10, 15, 19, 27, 34, 47, 59, 79, 99, 130, 162, 209, 259, 330, 407, 512, 628, 783, 956, 1181, 1435, 1760, 2129, 2594, 3124, 3784, 4539, 5468, 6534, 7834, 9327, 11132, 13208, 15701, 18568, 21989, 25923, 30592, 35960, 42297, 49579, 58139, 67967

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

a(n) is also the number of partitions of n with less than three 1's. - Geoffrey Critzer, Jun 20 2014

Crossrefs

Cf. A000041, A027336, A027338.

Column k=0 of A263232.

Column 3 of A175788.

Programs

Mathematica

nn=49;CoefficientList[Series[(1-x^3)Product[1/(1-x^i),{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Jun 20 2014 *)

PARI

a(n)=if(n<0,0,polcoeff((1-x^3)/eta(x+x*O(x^n)),n))

Formula

G.f.: (1-x^3) Product_{m>0} 1/(1-x^m).

a(n) = A000041(n) - A000041(n-3).

a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (4*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 3*Pi/(2*sqrt(6)))/sqrt(n) + (37/8 + 9/(2*Pi^2) + 1801*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016