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%I #12 Oct 04 2020 11:58:17
%S 1,1,6,59,810,14281,307566,7825859,229715130,7640988961,284037675966,
%T 11669182625099,525040651527210,25676859334384441,1356133254350401806,
%U 76928506160117877779,4664746297141400237850,301102611588796277314321,20613405033136513233790686,1491812049486032067219356699,113798761459974922574012320650
%N E.g.f. A(x) satisfies: A(x) = 1+x - cos(2*A(x) - x).
%H Paul D. Hanna, <a href="/A296169/b296169.txt">Table of n, a(n) for n = 1..300</a>
%F E.g.f. A(x) satisfies:
%F (1) A(x) = 1+x - cos(2*A(x) - x).
%F (2) A(x) = x + 2*sin(A(x) - x/2)^2.
%F (3) A(x) = x/2 + Series_Reversion( 2*x + 2*cos(2*x) - 2 ).
%F (4) A(x) = x/2 + Series_Reversion( 2*x - 4*sin(x)^2 ).
%F a(n) ~ 2^(n - 3/2) * 3^(n - 3/4) * n^(n-1) / ((Pi + 6*sqrt(3) - 12)^(n - 1/2) * exp(n)). - _Vaclav Kotesovec_, Oct 04 2020
%e E.g.f.: A(x) = x + x^2/2! + 6*x^3/3! + 59*x^4/4! + 810*x^5/5! + 14281*x^6/6! + 307566*x^7/7! + 7825859*x^8/8! + 229715130*x^9/9! + 7640988961*x^10/10! + ...
%e such that A(x) = 1+x - cos(2*A(x) - x).
%t terms = 21; A[_] = 0; Do[A[x_] = 1 + x - Cos[2*A[x] - x] + O[x]^(terms+1) // Normal, {terms+1}]; CoefficientList[A[x], x]*Range[0, terms]! // Rest (* _Jean-François Alcover_, Feb 05 2018 *)
%o (PARI) {a(n) = my(A = x/2 + serreverse(2*x - 4*sin(x +x*O(x^n))^2) ); n!*polcoeff(A,n)}
%o for(n=1,20,print1(a(n),", "))
%K nonn
%O 1,3
%A _Paul D. Hanna_, Feb 05 2018
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