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A026007
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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.
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78
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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
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OFFSET
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0,3
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COMMENTS
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Equals A000219: (1, 1, 3, 6, 13, 24, 48, 86, ...) convolved with the aerated version of the latter: (1, 0, 1, 0, 3, 0, 6, 0, 13, ...). - Gary W. Adamson, Jun 13 2009
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
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..10000
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, Graph - The asymptotic ratio
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.
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FORMULA
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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
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EXAMPLE
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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']
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI)
N=66; q='q+O('q^N);
gf= prod(n=1, N, (1+q^n)^n );
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */
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CROSSREFS
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Cf. A000009, A000027, A026741, A073592, A255528, A261562, A266857, A285223, A304040.
Cf. A000219. - Gary W. Adamson, Jun 13 2009
Cf. A027998, A248882, A248883, A248884.
Cf. A026011, A027346, A027906.
Column k=1 of A284992.
Sequence in context: A169826 A093065 A301596 * A032233 A026530 A336135
Adjacent sequences: A026004 A026005 A026006 * A026008 A026009 A026010
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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