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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
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, arXiv:2303.02240 [math.CO], 2023.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 18.
FORMULA
a(n) = (1/n)*Sum_{k=1..n} A078306(k)*a(n-k). - Vladeta Jovovic, Nov 22 2002
G.f.: Product_{m>=1} (1+x^m)^m. Weighout transform of natural numbers (A000027). Euler transform of A026741. - Franklin T. Adams-Watters, Mar 16 2006
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:
seq(a(n), n=0..45); # Alois P. Heinz, Aug 03 2013
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)
/* Joerg Arndt, Oct 06 2012 */
CROSSREFS
Cf. A000219. - Gary W. Adamson, Jun 13 2009
Column k=1 of A284992.
Sequence in context: A169826 A093065 A301596 * A032233 A026530 A336135
KEYWORD
nonn,nice
AUTHOR
STATUS
approved

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Last modified February 26 12:26 EST 2024. Contains 370352 sequences. (Running on oeis4.)