A100405 Number of partitions of n where every part appears more than two times.

1, 0, 0, 1, 1, 1, 2, 1, 2, 3, 3, 3, 7, 5, 6, 11, 10, 10, 17, 15, 20, 26, 25, 29, 44, 41, 47, 63, 67, 72, 99, 97, 114, 143, 148, 168, 216, 216, 248, 306, 328, 358, 443, 462, 527, 629, 665, 739, 898, 936, 1055, 1238, 1330, 1465, 1727, 1837, 2055, 2366, 2543, 2808, 3274

Offset: 0

Author

Vladeta Jovovic, Jan 11 2005

Keywords

• easy
• nonn

Examples

			a(6)=2 because we have [2,2,2] and [1,1,1,1,1,1].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..5000 from Vaclav Kotesovec)

Crossrefs

Cf. A007690, A160974-A160990.

Cf. A266647, A266648, A266649, A266650.

Cf. A161294.

Programs

Maple

G:=product((1+x^(3*k)/(1-x^k)),k=1..30): Gser:=series(G,x=0,80): seq(coeff(Gser,x,n),n=0..70); # Emeric Deutsch, Aug 06 2005
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1), j=[0, $3..iquo(n, i)])))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, Aug 20 2019

Mathematica

nmax = 100; Rest[CoefficientList[Series[Product[1 + x^(3*k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 28 2015 *)

Formula

G.f.: Product_{k>0} (1+x^(3*k)/(1-x^k)). More generally, g.f. for number of partitions of n where every part appears more than m times is Product_{k>0} (1+x^((m+1)*k)/(1-x^k)).

a(n) ~ sqrt(Pi^2 + 6*c) * exp(sqrt((2*Pi^2/3 + 4*c)*n)) / (4*sqrt(3)*Pi*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-3*x)) dx = -0.77271248407593487127235205445116662610863126869049971822566... . - Vaclav Kotesovec, Jan 05 2016

Extensions

More terms from Emeric Deutsch, Aug 06 2005

a(0)=1 prepended by Alois P. Heinz, Aug 20 2019