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%I #11 Oct 03 2023 05:58:02
%S 1,3,18,130,1044,8946,80135,741312,7027515,67911855,666525630,
%T 6625647054,66570488901,674964968175,6897258376218,70961851119848,
%U 734455079297433,7641851681095236,79886815507105175,838655487787502616,8837797224686207976,93454820274339167191
%N G.f.: Series reversion of x/P(x)^3 where P(x) is the g.f. for Partition numbers (A000041).
%F G.f. A(x) satisfies:
%F (1) A(x) = x/Product_{n>=1} (1 - A(x)^n)^3 ;
%F (2) A(x) = x/Sum_{n>=0} (-1)^n*(2n+1)*A(x)^(n(n+1)/2).
%F G.f.: A(x) = Series_Reversion(x*eta(x)^3) where eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
%F Self-convolution cube of A171804 (with offset).
%F a(n) ~ c * d^n / n^(3/2), where d = 11.34340769381039824727582112969136186... and c = 0.05972244738388663765328174469956... - _Vaclav Kotesovec_, Nov 11 2017
%e G.f.: A(x) = x + 3*x^2 + 18*x^3 + 130*x^4 + 1044*x^5 + 8946*x^6 +...
%e where Series_Reversion(A(x)) = x/P(x)^3 = x*eta(x)^3 and
%e x*eta(x)^3 = x - 3*x^2 + 5*x^4 - 7*x^7 + 9*x^11 - 11*x^16 + 13*x^22 +...
%t InverseSeries[x QPochhammer[x]^3 + O[x]^30][[3]] (* _Vladimir Reshetnikov_, Nov 21 2016 *)
%t (* Calculation of constants {d,c}: *) eq = FindRoot[{r/QPochhammer[s]^3 == s, 1/s + 3*(s/r)^(1/3)*Derivative[0, 1][QPochhammer][s, s] == (3*(Log[1 - s] + QPolyGamma[0, 1, s]))/(s*Log[s])}, {r, 1/10}, {s, 1/8}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[r*(-1 + s)*s^2*(Log[s]^2/(6*Pi*(r*(-4*s*ArcTanh[1 - 2*s] + Log[1 - s]*(2 + 3*(-1 + s)*Log[1 - s] + Log[s] - s*Log[s])) - (-1 + s)*(-3*r*QPolyGamma[0, 1, s]^2 + r*QPolyGamma[1, 1, s] + QPolyGamma[0, 1, s]*(r*(2 - 6*Log[1 - s] + Log[s]) + 6*(r/s)^(2/3)*s^2*Log[s]* Derivative[0, 1][QPochhammer][s, s]) + s*Log[s]*((r/s)^(1/3)*s*(6*(r/s)^(1/3) * Log[1 - s] * Derivative[0, 1][QPochhammer][s, s] - 4*s*Log[s] * Derivative[0, 1][QPochhammer][s, s]^2 + (r/s)^(1/3)*s*Log[s]* Derivative[0, 2][QPochhammer][s, s]) - 2*r*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(serreverse(x*eta(x+x*O(x^n))^3),n)}
%Y Cf. A109085, A171802, A171803, A171804, A000041, A007312, A010815, A010816.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Dec 20 2009
%E More terms from _Vladimir Reshetnikov_, Nov 21 2016
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