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%I #68 Jan 24 2024 08:01:49
%S 1,1,2,5,8,16,28,49,83,142,235,385,627,1004,1599,2521,3940,6111,9421,
%T 14409,21916,33134,49808,74484,110837,164132,241960,355169,519158,
%U 755894,1096411,1584519,2281926,3275276,4685731,6682699,9501979,13471239,19044780,26850921,37756561,52955699
%N 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.
%C 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
%H Alois P. Heinz, <a href="/A026007/b026007.txt">Table of n, a(n) for n = 0..10000</a>
%H Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, <a href="https://arxiv.org/abs/2303.02240">A unified treatment of families of partition functions</a>, arXiv:2303.02240 [math.CO], 2023.
%H Vaclav Kotesovec, <a href="/A026007/a026007.jpg">Graph - The asymptotic ratio</a>
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 18.
%F a(n) = (1/n)*Sum_{k=1..n} A078306(k)*a(n-k). - _Vladeta Jovovic_, Nov 22 2002
%F 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
%F 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
%e For n = 4, we have 8 partitions
%e 01: [4]
%e 02: [4']
%e 03: [4'']
%e 04: [4''']
%e 05: [3, 1]
%e 06: [3', 1]
%e 07: [3'', 1]
%e 08: [2, 2']
%p with(numtheory):
%p b:= proc(n) option remember;
%p add((-1)^(n/d+1)*d^2, d=divisors(n))
%p end:
%p a:= proc(n) option remember;
%p `if`(n=0, 1, add(b(k)*a(n-k), k=1..n)/n)
%p end:
%p seq(a(n), n=0..45); # _Alois P. Heinz_, Aug 03 2013
%t 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_ *)
%t 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 *)
%o (PARI)
%o N=66; q='q+O('q^N);
%o gf= prod(n=1,N, (1+q^n)^n );
%o Vec(gf)
%o /* _Joerg Arndt_, Oct 06 2012 */
%Y Cf. A000009, A000027, A026741, A073592, A255528, A261562, A266857, A285223, A304040.
%Y Cf. A000219. - _Gary W. Adamson_, Jun 13 2009
%Y Cf. A027998, A248882, A248883, A248884.
%Y Cf. A026011, A027346, A027906.
%Y Column k=1 of A284992.
%K nonn,nice
%O 0,3
%A _N. J. A. Sloane_
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