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%I #7 Oct 06 2020 09:07:46
%S 1,1,10,165,3620,96600,2996586,105222740,4110953640,176563668420,
%T 8262487393200,418357209052255,22790291724038580,1329373540730613885,
%U 82689132241752729840,5464831115387029530825,382492359049214499350160,28269306654212844076331775,2200386909130514596330158960,179936864840741390285785502825,15424543660650808111807757322400
%N G.f. A(x) satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^4)].
%H Vaclav Kotesovec, <a href="/A305604/b305604.txt">Table of n, a(n) for n = 0..360</a>
%F O.g.f. A(x) satisfies:
%F (1) [x^n] exp( n * Integral A(x)^4 dx ) * (n + 1 - A(x)) = 0 for n > 0.
%F (2) A(x) = 1 + x*A(x)^3*(A(x) + 4*x*A'(x))/(1 - x*A(x)^4)^2.
%F a(n) ~ c * 4^n * n^(5/4) * n!, where c = 0.147639333661398142298711... - _Vaclav Kotesovec_, Oct 06 2020
%e G.f.: A(x) = 1 + x + 10*x^2 + 165*x^3 + 3620*x^4 + 96600*x^5 + 2996586*x^6 + 105222740*x^7 + 4110953640*x^8 + 176563668420*x^9 + ...
%e such that A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^4)].
%e RELATED SERIES.
%e A(x)^4 = 1 + 4*x + 46*x^2 + 784*x^3 + 17181*x^4 + 452860*x^5 + 13831594*x^6 + 478200572*x^7 + 18418253542*x^8 + 781180290204*x^9 + ...
%e 1/(1 - x*A(x)^4) = 1 + x + 5*x^2 + 55*x^3 + 905*x^4 + 19320*x^5 + 499431*x^6 + 15031820*x^7 + 513869205*x^8 + 19618185380*x^9 + ...
%e A'(x)/A(x) = 1 + 19*x + 466*x^2 + 13659*x^3 + 457926*x^4 + 17142730*x^5 + 706064549*x^6 + 31677960427*x^7 + 1537022113117*x^8 + ...
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1 - x*A^4+x*O(x^n)))); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*intformal(Ser(A)^4) ) * ((m-1) + 1 - Ser(A)) )[m] ); A[n+1]}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A305602, A305603, A209881, A305110.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 05 2018
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