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%I #15 Sep 26 2023 12:49:46
%S 1,1,4,20,115,714,4669,31671,220800,1572395,11389059,83642650,
%T 621400794,4661706035,35264616260,268700873765,2060348179869,
%U 15886552304352,123102352038195,958128272163860,7487015421267228,58715989507106041
%N G.f. satisfies: A(x) = P(x*A(x)^2) where A(x/P(x)^2) = P(x) is the g.f. for Partition numbers (A000041).
%H Vaclav Kotesovec, <a href="/A171802/b171802.txt">Table of n, a(n) for n = 0..500</a>
%F G.f. A(x) satisfies [_Paul D. Hanna_, Nov 24 2012]:
%F (1) A(x) = (1/x)*series_reversion(x*eta(x)^2).
%F (2) A(x) = 1 / Product_{n>=1} (1 - x^n*A(x)^(2*n)).
%F (3) A(x) = Sum_{n>=0} x^n*A(x)^(2*n) / Product_{k=1..n} (1-x^k*A(x)^(2*k)).
%F (4) A(x) = Sum_{n>=0} (x*A(x)^2)^(n^2) / Product_{k=1..n} (1-x^k*A(x)^(2*k))^2.
%F (5) A(x) = exp( Sum_{n>=1} (x^n/n) * A(x)^(2*n)/(1 - x^n*A(x)^(2*n)) ).
%F a(n) ~ c * d^n / n^(3/2), where d = 8.42516721063251541777601555584151410936... and c = 0.2128745515668564974075326286129891378270... - _Vaclav Kotesovec_, May 13 2018
%e G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 115*x^4 + 714*x^5 +...
%e G.f. satisfies A(x/P(x)^2) = P(x) where:
%e P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 +...
%e and x/P(x)^2 = x - 2*x^2 - x^3 + 2*x^4 + x^5 + 2*x^6 - 2*x^7 - 2*x^9 +...
%t nmax = 20; A[_] = 0; Do[A[x_] = 1/Product[1 - x^k*A[x]^(2*k), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* _Vaclav Kotesovec_, Sep 26 2023 *)
%t (* Calculation of constants {d,c}: *) eq = FindRoot[{s*QPochhammer[r*s^2] == 1, 1/s + 2*r*s^2*Derivative[0, 1][QPochhammer][r*s^2, r*s^2] == (2*(Log[1 - r*s^2] + QPolyGamma[0, 1, r*s^2]))/(s* Log[r*s^2])}, {r, 1/8}, {s, 1}, WorkingPrecision -> 1000]; {N[1/r /. eq, 100], val = -s* Log[r*s^2]*(Sqrt[1 - r*s^2]/ Sqrt[4*Pi*(16*r*s^2*ArcTanh[1 - 2*r*s^2] + (1 - r*s^2)*(Log[r*s^2] - 2*Log[1 - r*s^2])*(3*Log[r*s^2] - 2*Log[1 - r*s^2]) - 8*Log[1 - r*s^2] + 8*(1 - r*s^2)*(-1 + 2*ArcTanh[1 - 2*r*s^2]) * QPolyGamma[0, 1, r*s^2] + 4*(1 - r*s^2)*QPolyGamma[0, 1, r*s^2]^2 - 4*(1 - r*s^2)*(QPolyGamma[1, 1, r*s^2] + r*s^2*Log[r*s^2]*(r*s^3*Log[r*s^2]* Derivative[0, 2][QPochhammer][r*s^2, r*s^2] - 2* Derivative[0, 0, 1][QPolyGamma][0, 1, r*s^2])))]) /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* _Vaclav Kotesovec_, Sep 26 2023 *)
%o (PARI) a(n)=polcoeff((1/x*serreverse(x*eta(x+x*O(x^n))^2))^(1/2), n)
%Y Cf. A171803, A171804, A171805, A109085, A000041, A304444, A366026.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 19 2009
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