A181531 Number of partitions of n with no part equal to 1 or 3.

1, 0, 1, 0, 2, 1, 3, 2, 5, 4, 8, 7, 13, 12, 20, 20, 31, 32, 47, 50, 71, 77, 105, 116, 155, 173, 225, 254, 325, 369, 465, 530, 660, 755, 929, 1066, 1300, 1493, 1805, 2076, 2493, 2867, 3421, 3934, 4669, 5368, 6337, 7282, 8560, 9828, 11505, 13198, 15394, 17641, 20507, 23475

Offset

0,5

Links

Table of n, a(n) for n=0..55.

Formula

a(n) = p(n) - p(n-1) - p(n-3) + p(n-4), where p(n) = A000041(n).

G.f.: (1-x-x^3+x^4) / Product_{m>=1} (1-x^m).

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^2 / (8*sqrt(3)*n^2). - Vaclav Kotesovec, Jun 02 2018

Example

a(6)=3 because we have [2,2,2], [2,4], and [6].

Maple

with(combinat): p := proc (n) options operator, arrow: numbpart(n) end proc: 1, 0, 1, 0, 2, seq(p(n)-p(n-1)-p(n-3)+p(n-4), n = 5 .. 55);
G := (1-x)*(1-x^3)/(product(1-x^j, j = 1 .. 65)): Gser := series(G, x = 0, 60): seq(coeff(Gser, x, n), n = 0 .. 55);

Mathematica

Table[Count[IntegerPartitions[n], _?(FreeQ[#, 1]&&FreeQ[#, 3]&)], {n, 0, 60}] (* Harvey P. Dale, Feb 25 2015 *)

Crossrefs

Cf. A000041.

Sequence in context: A005916 A034392 A361392 * A034393 A068932 A151533

Adjacent sequences: A181528 A181529 A181530 * A181532 A181533 A181534

Keyword

nonn

Author

Shanzhen Gao, Oct 28 2010

Status

approved