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%I #23 Oct 15 2020 13:26:33
%S 1,1,3,16,118,1050,10509,113892,1307043,15661024,194075098,2470848492,
%T 32161635070,426440290744,5743575712131,78405535427220,
%U 1082876597440146,15109514661352482,212736976140479073,3019422091269739704,43164665664066028062,621078277521084894978,8989001884449529431990,130795752983608734209604,1912460927749734257739153,28088780052768915388505436,414247711043291214286003410
%N O.g.f. A(x) satisfies: A(x) = 1 + Integral (x/A(x))' / (x/A(x)^4)' dx.
%H Paul D. Hanna, <a href="/A302701/b302701.txt">Table of n, a(n) for n = 0..400</a>
%F O.g.f. A(x) satisfies:
%F (1) A(x) = 1 + Integral (x/A(x))' / (x/A(x)^4)' dx.
%F (2) A(x) = 1 + Integral A(x)^3 * (A(x) - x*A'(x)) / (A(x) - 4*x*A'(x)) dx.
%F (3) A(x) = 1 + Integral A(x) * (1 + x*A(x)^2 - sqrt(1 - 14*x*A(x)^2 + x^2*A(x)^4) )/(8*x) dx.
%F (4) 0 = A(x)^4 - A(x)*(1 + x*A(x)^2)*A'(x) + 4*x*A'(x)^2.
%F a(n) ~ 3^(2/3) * (1240209 - 716035*sqrt(3))^(1/6) * 2^((4*n - 5)/3) * (3 + 2*sqrt(3))^n / (sqrt(Pi) * n^(5/2)). - _Vaclav Kotesovec_, Oct 14 2020
%e G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 118*x^4 + 1050*x^5 + 10509*x^6 + 113892*x^7 + 1307043*x^8 + 15661024*x^9 + 194075098*x^10 + ...
%e RELATED SERIES.
%e (x/A(x))' / (x/A(x)^4)' = 1 + 6*x + 48*x^2 + 472*x^3 + 5250*x^4 + 63054*x^5 + 797244*x^6 + 10456344*x^7 + 140949216*x^8 + 1940750980*x^9 + ...
%e which equals A'(x).
%e The logarithmic derivative of the g.f. begins:
%e A'(x)/A(x) = 1 + 5*x + 40*x^2 + 401*x^3 + 4531*x^4 + 55040*x^5 + 701716*x^6 + 9261257*x^7 + 125449600*x^8 + 1734071855*x^9 + 24362189248*x^10 + ...
%e which equals (1 + x*A(x)^2 - sqrt(1 - 14*x*A(x)^2 + x^2*A(x)^4))/(8*x).
%t nmax = 30; A = 1; Do[A = 1 + Integrate[D[x/A, x]/D[x/A^4, x], x] + O[x]^nmax, nmax]; CoefficientList[A, x] (* _Vaclav Kotesovec_, Oct 15 2020 *)
%o (PARI) {a(n) = my(A=1); for(i=1, n, A = 1 + intformal( (x/A)'/(x/A^4 +x*O(x^n))' ); ); polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A302704, A302705, A303064, A338163, A338187, A338188, A338193, A338194.
%K nonn,nice
%O 0,3
%A _Paul D. Hanna_, Apr 19 2018