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%I #19 Mar 16 2024 08:47:39
%S 1,1,3,19,181,2121,28035,401199,6076494,96043696,1569407902,
%T 26338955412,451829668906,7894022242204,140075248932330,
%U 2518908613782600,45823514062061016,842108582257569081,15614889082228858722,291858857158743005901,5494258022591894716440,104097462455681871262881,1983820645046435115347970,38007365345354099879246673
%N O.g.f. A(x) satisfies: A(x) = 1 + Integral (x/A(x)^4)' / (x/A(x)^7)' dx.
%H Vaclav Kotesovec, <a href="/A303064/b303064.txt">Table of n, a(n) for n = 0..500</a> (terms 0..200 from Paul D. Hanna)
%F O.g.f. A(x) satisfies:
%F (1) A(x) = 1 + Integral (x/A(x)^4)' / (x/A(x)^7)' dx.
%F (2) A(x) = 1 + Integral A(x)^3 * (A(x) - 4*x*A'(x)) / (A(x) - 7*x*A'(x)) dx.
%F (3) A(x) = 1 + Integral A(x) * (1 + 4*x*A(x)^2 - sqrt(1 - 20*x*A(x)^2 + 16*x^2*A(x)^4) )/(14*x) dx.
%F (4) 0 = A(x)^4 - A(x)*(1 + 4*x*A(x)^2)*A'(x) + 7*x*A'(x)^2.
%F a(n) ~ c * d^n / n^(5/2), where d = 2^(5/3)*(3/7)^(4/9)*(432893979 + 94465211*sqrt(21))^(1/9) = 21.43962319881971664190505405921680468600... and c = 0.022570265358175200394042178896826753964244... - _Vaclav Kotesovec_, Oct 14 2020, updated Mar 16 2024
%e G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 181*x^4 + 2121*x^5 + 28035*x^6 + 401199*x^7 + 6076494*x^8 + 96043696*x^9 + 1569407902*x^10 + ...
%e RELATED SERIES.
%e (x/A(x)^4)' / (x/A(x)^7)' = 1 + 6*x + 57*x^2 + 724*x^3 + 10605*x^4 + 168210*x^5 + 2808393*x^6 + 48611952*x^7 + 864393264*x^8 + 15694079020*x^9 + ...
%e which equals A'(x).
%e The logarithmic derivative of the g.f. begins:
%e A'(x)/A(x) = 1 + 5*x + 49*x^2 + 641*x^3 + 9541*x^4 + 152789*x^5 + 2567293*x^6 + 44643689*x^7 + 796602019*x^8 + 14502820745*x^9 + ...
%e which equals (1 + 4*x*A(x)^2 - sqrt(1 - 20*x*A(x)^2 + 16*x^2*A(x)^4))/(14*x).
%t nmax = 25; A = 1; Do[A = 1 + Integrate[D[x/A^4, x]/D[x/A^7, 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^4)'/(x/A^7 +x*O(x^n))' ); ); polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A302701, A302704, A302705, A338163, A338187, A338188, A338193, A338194.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 19 2018
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