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%I #31 Mar 26 2018 13:06:09
%S 1,2,6,22,94,474,2974,24630,271710,3799570,63378806,1208997078,
%T 25736584670,602485683530,15356903176110,423032451327510,
%U 12518043710674878,395909541133928226,13325077980379707238,475466006418129789206,17926802213221278261726,712095926927360739006522,29722097317161256669118142,1300445348644716445771904502
%N O.g.f. A(x) satisfies: A(x) = x * (1 + 3*x*A'(x)) / (1 + x*A'(x)).
%C If G(x) = x*(1 + r*x*G'(x)) / (1 + x*G'(x)), then G(x) has negative coefficients if r < t, and consists entirely of nonnegative coefficients if r > t, where t = 2.8453449032025472172778433620905570976610361149... (A301389).
%C O.g.f. equals the logarithm of the e.g.f. of A301386.
%C The e.g.f. G(x) of A301386 satisfies: [x^n] G(x)^(-n) = (2*n - 3) * [x^(n-1)] G(x)^(-n) for n>=1.
%H Paul D. Hanna, <a href="/A301385/b301385.txt">Table of n, a(n) for n = 1..300</a>
%F O.g.f. A(x) satisfies: [x^n] exp( -n * A(x) ) = (2*n - 3) * [x^(n-1)] exp( -n * A(x) ) for n>=1.
%F From _Vaclav Kotesovec_, Mar 20 2018: (Start)
%F a(n) ~ c * 2^n * n! / n^2, where c = 0.0618315205229178422646235585879521967924163...
%F a(n) ~ c * 2^n * n^(n - 3/2) / exp(n), where c = 0.15498863760617284891466946263730170095444214... (End)
%e G.f.: A(x) = x + 2*x^2 + 6*x^3 + 22*x^4 + 94*x^5 + 474*x^6 + 2974*x^7 + 24630*x^8 + 271710*x^9 + 3799570*x^10 + ...
%e where
%e A(x) = x*(1 + 3*x*A'(x)) / (1 + x*A'(x)).
%e RELATED SERIES.
%e A'(x) = 1 + 4*x + 18*x^2 + 88*x^3 + 470*x^4 + 2844*x^5 + 20818*x^6 + 197040*x^7 + 2445390*x^8 + 37995700*x^9 + ...
%e exp(A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 745*x^4/4! + 16001*x^5/5! + 472621*x^6/6! + 19659025*x^7/7! + 1211940689*x^8/8! + ... + A301386*x^n/n! + ...
%t Rest[CoefficientList[AsymptoticDSolveValue[{A[x] == x*(1 + 3*x*A'[x])/(1 + x*A'[x]), A[1] == 1}, A[x], {x, 0, 20}], x]] (* Requires Mathematica version 11.3 or later *) (* _Vaclav Kotesovec_, Mar 20 2018 *)
%o (PARI) {a(n) = my(A=x); for(i=0,n, A = x*(1 + 3*x*A')/(1 +x*A' +x*O(x^n)) ); polcoeff(A,n)}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A301386, A301388, A301389, A300736, A300987.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Mar 20 2018
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