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A100405
Number of partitions of n where every part appears more than two times.
19
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
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