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

0,7

Links

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

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

Example

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

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 *)

Crossrefs

Cf. A007690, A160974-A160990.

Cf. A266647, A266648, A266649, A266650.

Sequence in context: A161078 A161294 A161269 * A081366 A129636 A242443

Adjacent sequences: A100402 A100403 A100404 * A100406 A100407 A100408

Keyword

easy,nonn

Author

Vladeta Jovovic, Jan 11 2005

Extensions

More terms from Emeric Deutsch, Aug 06 2005

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

Status

approved