A027343 Number of partitions of n that do not contain 9 as a part.

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 54, 74, 96, 128, 165, 216, 275, 355, 448, 571, 715, 901, 1120, 1399, 1727, 2139, 2625, 3228, 3938, 4812, 5840, 7094, 8568, 10352, 12447, 14967, 17919, 21450, 25581, 30496, 36234, 43031, 50951, 60292

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Crossrefs

9th column of A175788. Cf. A000041, A027336, A027337-A027344.

Programs

Maple

A41:= n-> `if`(n<0, 0, combinat[numbpart](n)):
a:= n-> A41(n) -A41(n-9):
seq(a(n), n=0..50);

Mathematica

Table[Count[IntegerPartitions[n],?(FreeQ[#,9]&)],{n,0,50}] (* _Harvey P. Dale, Oct 02 2022 *)

Formula

G.f.: (1-x^9) Product_{m>0} 1/(1-x^m).

a(n) = A000041(n)-A000041(n-9).

a(n) ~ 3*Pi * exp(sqrt(2*n/3)*Pi) / (4*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 9*Pi/(2*sqrt(6)))/sqrt(n) + (109/8 + 9/(2*Pi^2) + 15769*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

Extensions

Edited by Alois P. Heinz, Dec 04 2010