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A027998 Expansion of Product_{m>=1} (1+q^m)^(m^2).

%I A027998 #48 Sep 08 2022 08:44:50
%S A027998 1,1,4,13,31,83,201,487,1141,2641,5972,13309,29248,63360,135688,
%T A027998 287197,601629,1247909,2565037,5226816,10565132,21192569,42202909,
%U A027998 83466925,163999684,320230999,621579965,1199659836,2302765961,4397132933,8354234552,15795913477
%N A027998 Expansion of Product_{m>=1} (1+q^m)^(m^2).
%C A027998 In general, if g.f. = Product_{k>=1} (1 + x^k)^(c2*k^2 + c1*k + c0) and c2 > 0, then a(n) ~ exp(2*Pi/3 * (14*c2/15)^(1/4) * n^(3/4) + 3*c1 * Zeta(3) / Pi^2 * sqrt(15*n/(14*c2)) + (Pi * c0 * (5/(14*c2))^(1/4) / (2*3^(3/4)) - 9*c1^2 * Zeta(3)^2 * (15/(14*c2))^(5/4) / Pi^5) * n^(1/4) + 2025 * c1^3 * Zeta(3)^3 / (49 * c2^2 * Pi^8) - 15*c0*c1*Zeta(3) / (28*c2 * Pi^2)) * ((7*c2)/15)^(1/8) / (2^(15/8 + c0/2 + c1/12) * n^(5/8)). - _Vaclav Kotesovec_, Nov 08 2017
%H A027998 Seiichi Manyama, <a href="/A027998/b027998.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%H A027998 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. 22.
%F A027998 a(n) = 1/n*Sum_{k=1..n} A078307(k)*a(n-k). - _Vladeta Jovovic_, Nov 22 2002
%F A027998 a(n) ~ 7^(1/8) * exp(2/3 * Pi * (14/15)^(1/4) * n^(3/4)) / (2^(15/8) * 15^(1/8) * n^(5/8)). - _Vaclav Kotesovec_, Mar 05 2015
%F A027998 G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + x^k)/(k*(1 - x^k)^3)). - _Ilya Gutkovskiy_, May 30 2018
%p A027998 with(numtheory):
%p A027998 b:= proc(n) option remember;
%p A027998       add((-1)^(n/d+1)*d^3, d=divisors(n))
%p A027998     end:
%p A027998 a:= proc(n) option remember;
%p A027998       `if`(n=0, 1, add(b(k)*a(n-k), k=1..n)/n)
%p A027998     end:
%p A027998 seq(a(n), n=0..35);  # _Alois P. Heinz_, Aug 03 2013
%t A027998 a[0] = 1; a[n_] := a[n] = 1/n*Sum[Sum[(-1)^(k/d+1)*d^3, {d, Divisors[k]}]*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 31} ] (* _Jean-François Alcover_, Jan 17 2014, after _Vladeta Jovovic_ *)
%t A027998 nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^2),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Mar 05 2015 *)
%o A027998 (PARI) x = 'x + O('x ^ 50); Vec(prod(k=1, 50, (1 + x^k)^(k^2))) \\ _Indranil Ghosh_, Apr 05 2017
%o A027998 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^2: k in [1..m]]) )); // _G. C. Greubel_, Oct 31 2018
%Y A027998 Cf. A026007, A248882, A248883, A248884, A285224, A304049.
%Y A027998 Column k=2 of A284992.
%K A027998 nonn
%O A027998 0,3
%A A027998 _N. J. A. Sloane_