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%I #10 Oct 03 2023 05:28:53
%S 1,2,9,48,286,1818,12086,82992,584079,4190738,30539814,225426240,
%T 1681904909,12663614266,96099303213,734250983952,5643749482600,
%U 43610375803722,338578974873523,2639771240159904,20659895819582337
%N G.f. satisfies: A(x) = P(x*A(x))^2 where A(x/P(x)^2) = P(x)^2 and P(x) is the g.f. for Partition numbers (A000041).
%F G.f. satisfies: A(x) = 1/Product_{n>=1} (1 - A(x)^n)^2.
%F G.f.: A(x) = Series_Reversion(x*eta(x)^2) where eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
%F Self-convolution of A171802.
%F From _Vaclav Kotesovec_, Nov 11 2017: (Start)
%F a(n) ~ c * d^n / n^(3/2), where
%F d = 8.4251672106325154177760155558415141093613298032469849432733825... and
%F c = 0.6057593757525562292332998445991464666128350560350232598293... (End)
%e G.f.: A(x) = 1 + 2*x + 9*x^2 + 48*x^3 + 286*x^4 + 1818*x^5 +...
%e A(x/P(x)^2) = P(x)^2 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 P(x)^2 = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 110*x^7 +...
%t nmax = 25; Rest[CoefficientList[InverseSeries[Series[x*Product[(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}]], x]] (* _Vaclav Kotesovec_, Nov 11 2017 *)
%t nmax = 30; A[_] = 0; Do[A[x_] = x/Product[(1 - A[x]^k)^2, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x]/x, x] (* _Vaclav Kotesovec_, Oct 03 2023 *)
%t (* Calculation of constants {d,c}: *) eq = FindRoot[{r/QPochhammer[s]^2 == s, 1/s + 2*Sqrt[s/r]*Derivative[0, 1][QPochhammer][s, s] == (2*(Log[1 - s] + QPolyGamma[0, 1, s]))/(s*Log[s])}, {r, 1/8}, {s, 1/4}, WorkingPrecision -> 1200]; {N[1/r /. eq, 120], val = -s*Log[s]*Sqrt[(-1 + s)/(Pi*r*(r*(-8*s*Log[-1 + 1/s] + 4*(-1 + s)*Log[1 - s]^2 + 3*(-1 + s)*Log[s]^2 + 8*Log[1 - s]*(1 + Log[s] - s*Log[s])) + 8*r*(-1 + s)*(-1 + Log[-1 + 1/s])* QPolyGamma[0, 1, s] + 4*r*(-1 + s)*QPolyGamma[0, 1, s]^2 - 4*r*(-1 + s)*QPolyGamma[1, 1, s] - 4*Sqrt[r]*(-1 + s)*s^(5/2)*Log[s]^2* Derivative[0, 2][QPochhammer][s, s] + 8*r*(-1 + s)*s*Log[s]* Derivative[0, 0, 1][QPolyGamma][0, 1, s]))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* _Vaclav Kotesovec_, Oct 03 2023 *)
%o (PARI) a(n)=polcoeff(1/x*serreverse(x*eta(x+x*O(x^n))^2), n)
%Y Cf. A171802, A171804, A171805, A109085, A000041, A304444.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 19 2009
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