|
|
A026007
|
|
Expansion of Product_{m>=1} (1 + q^m)^m; number of partitions of n into distinct parts, where n different parts of size n are available.
|
|
79
|
|
|
1, 1, 2, 5, 8, 16, 28, 49, 83, 142, 235, 385, 627, 1004, 1599, 2521, 3940, 6111, 9421, 14409, 21916, 33134, 49808, 74484, 110837, 164132, 241960, 355169, 519158, 755894, 1096411, 1584519, 2281926, 3275276, 4685731, 6682699, 9501979, 13471239, 19044780, 26850921, 37756561, 52955699
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
In general, for t > 0, if g.f. = Product_{m>=1} (1 + t*q^m)^m then a(n) ~ c^(1/6) * exp(3^(2/3) * c^(1/3) * n^(2/3) / 2) / (3^(2/3) * (t+1)^(1/12) * sqrt(2*Pi) * n^(2/3)), where c = Pi^2*log(t) + log(t)^3 - 6*polylog(3, -1/t). - Vaclav Kotesovec, Jan 04 2016
|
|
LINKS
|
|
|
FORMULA
|
a(n) ~ zeta(3)^(1/6) * exp((3/2)^(4/3) * zeta(3)^(1/3) * n^(2/3)) / (2^(3/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)), where zeta(3) = A002117. - Vaclav Kotesovec, Mar 05 2015
|
|
EXAMPLE
|
For n = 4, we have 8 partitions
01: [4]
02: [4']
03: [4'']
04: [4''']
05: [3, 1]
06: [3', 1]
07: [3'', 1]
08: [2, 2']
|
|
MAPLE
|
with(numtheory):
b:= proc(n) option remember;
add((-1)^(n/d+1)*d^2, d=divisors(n))
end:
a:= proc(n) option remember;
`if`(n=0, 1, add(b(k)*a(n-k), k=1..n)/n)
end:
|
|
MATHEMATICA
|
a[n_] := a[n] = 1/n*Sum[Sum[(-1)^(k/d+1)*d^2, {d, Divisors[k]}]*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 41}] (* Jean-François Alcover, Apr 17 2014, after Vladeta Jovovic *)
nmax=50; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*x^k/(k*(1-x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 28 2015 *)
|
|
PROG
|
(PARI)
N=66; q='q+O('q^N);
gf= prod(n=1, N, (1+q^n)^n );
Vec(gf)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,nice,changed
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|