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A192619 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x)^(1/2))^2/(1 - x^n*A(x)^(1/2))^2.

%I A192619 #14 Jun 30 2025 05:04:53
%S A192619 1,4,20,104,556,3048,17064,97216,562036,3289836,19461448,116178600,
%T A192619 699045176,4235292680,25816944176,158223753376,974389668364,
%U A192619 6026623271840,37420762694588,233179517592232,1457706542138344,9139698522931008
%N A192619 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x)^(1/2))^2/(1 - x^n*A(x)^(1/2))^2.
%H A192619 Vaclav Kotesovec, <a href="/A192619/b192619.txt">Table of n, a(n) for n = 0..250</a>
%F A192619 Self-convolution of A190862.
%F A192619 a(n) ~ c * d^n / n^(3/2), where d = 6.693428901114353533300254329706934045134... and c = 4.7342954578062245798237099751798009... - _Vaclav Kotesovec_, Jun 30 2025
%e A192619 G.f.: A(x) = 1 + 4*x + 20*x^2 + 104*x^3 + 556*x^4 + 3048*x^5 +...
%e A192619 The g.f. A = A(x) satisfies:
%e A192619 A = (1+x*A^(1/2))^2/(1-x*A^(1/2))^2 * (1+x^2*A^(1/2))^2/(1-x^2*A^(1/2))^2 * (1+x^3*A^(1/2))^2/(1-x^3*A^(1/2))^2 *...
%t A192619 (* Calculation of constants {d,c}: *) Chop[{1/r, (1 - s)*s*Log[r]* Sqrt[(2*r*(QPochhammer[Sqrt[s], r]* Derivative[0, 1][QPochhammer][-Sqrt[s], r] - QPochhammer[-Sqrt[s], r]* Derivative[0, 1][QPochhammer][Sqrt[s], r]))/ (Pi* QPochhammer[s, r^2]*(-2*Sqrt[s]*(1 + s)*Log[r]^2 + (-1 + s)^2* QPolyGamma[1, Log[s]/Log[r^2], r] - (-1 + s)^2* QPolyGamma[1, (2*I*Pi + Log[s])/Log[r^2], r]))]} /. FindRoot[{((-1 + Sqrt[s])^2* QPochhammer[-Sqrt[s], r]^2)/((1 + Sqrt[s])^2* QPochhammer[Sqrt[s], r]^2) == s, (-1 - 2*Sqrt[s] + s)/(-1 + s) + (QPolyGamma[0, Log[-Sqrt[s]]/Log[r], r] - QPolyGamma[0, Log[s]/(2*Log[r]), r])/ Log[r] == 0}, {r, 1/6}, {s, 2}, WorkingPrecision -> 120]] (* _Vaclav Kotesovec_, Jun 30 2025 *)
%o A192619 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1+x^m*A^(1/2))^2/(1-x^m*A^(1/2)+x*O(x^n))^2));polcoeff(A, n)}
%o A192619 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+sum(m=1, n, x^m*A^(m/2)*prod(k=1, m,(1+x^(k-1))/(1-x^k+x*O(x^n)))))^2); polcoeff(A, n)}
%Y A192619 Cf. A190862.
%K A192619 nonn
%O A192619 0,2
%A A192619 _Paul D. Hanna_, Jul 06 2011