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%I #16 Oct 27 2024 06:16:01
%S 1,1,2,10,100,1556,33016,888952,29035280,1115554960,49300214176,
%T 2463859486496,137403573562432,8459184183342400,569861708317147520,
%U 41697486853043633536,3293243089832744386816,279234174032057551630592,25299360271944290704683520
%N E.g.f. A(x) satisfies: A(x) = 1 + Integral (x/A(x))' / (x/A(x)^2)' dx.
%H Vaclav Kotesovec, <a href="/A338193/b338193.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) ~ (1 + sqrt(2))^(2*n - 3/2) * n^(n-2) / (2^(3/4) * exp(n - 1 + 1/sqrt(2))).
%F E.g.f.: (8*x^2 * (x - 3 + sqrt(1 - 6*x + x^2))^3 / (3*x - 1 + sqrt(1 - 6*x + x^2)))^(1/4) * exp((1 + x - sqrt(1 - 6*x + x^2))/4)/2.
%F Recurrence: (2*n - 1)*a(n) = (12*n^2 - 35*n + 24)*a(n-1) - (n-1)*(2*n^2 - 5*n - 2)*a(n-2) + (n-4)*(n-2)*(n-1)*a(n-3). - _Vaclav Kotesovec_, Oct 26 2024
%F Conjecture: a(n) = f(0, n-1) for n > 0 with a(0) = 1 where f(n, m) = f(n-1, m) + m*(f(n, m-1) + f(n+1, m-1)) for n > 0, m > 0 with f(0, m) = f(0, m-1) + m*f(1, m-1) for m > 0, f(n, 0) = 1 for n >= 0. - _Mikhail Kurkov_, Oct 26 2024
%t nmax = 20; A = 1; Do[A = 1 + Integrate[D[x/A, x]/D[x/A^2, x], x] + O[x]^nmax, nmax]; CoefficientList[A, x] * Range[0, nmax - 1]!
%t nmax = 20; FullSimplify[CoefficientList[Series[(8*x^2 * (x - 3 + Sqrt[1 - 6*x + x^2])^3 / (3*x - 1 + Sqrt[1 - 6*x + x^2]))^(1/4) * E^((1 + x - Sqrt[1 - 6*x + x^2])/4)/2, {x, 0, nmax}], x] * Range[0, nmax]!]
%o (PARI) {a(n) = my(A=1); for(i=1, n, A = 1 + intformal( (x/A)'/(x/A^2 +x*O(x^n))' ); ); n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A302701, A303064, A338163, A338187, A338188, A338194.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Oct 15 2020