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A318004 E.g.f.: 2*cos(x) / (1 + sqrt(1 - 2*sin(2*x))). 1

%I

%S 1,1,3,23,249,3601,65163,1420103,36240369,1060638241,35030837523,

%T 1289122462583,52311218246889,2320745189970481,111753587921091483,

%U 5805372695984119463,323619830261141155809,19269740737912396000321,1220661620760214878827043,81966845938603736180310743,5815923258824904181135143129

%N E.g.f.: 2*cos(x) / (1 + sqrt(1 - 2*sin(2*x))).

%F E.g.f. A(x) satisfies:

%F (1a) A(x) = cos(x) + sin(x)*A(x)^2.

%F (1b) A(x) = cos(x) * Sum_{n>=0} binomial(2*n,n)/(n+1) * sin(2*x)^n/2^n.

%F (2a) A(x) = (1 - sqrt(1 - 2*sin(2*x))) / (2*sin(x)).

%F (2b) A(x) = 2*cos(x) / (1 + sqrt(1 - 2*sin(2*x))).

%F a(n) ~ (1 + sqrt(3)) * 2^(2*n - 3/2) * 3^(n - 1/4) * n^(n-1) / (exp(n) * Pi^(n - 1/2)). - _Vaclav Kotesovec_, Aug 21 2018

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 23*x^3/3! + 249*x^4/4! + 3601*x^5/5! + 65163*x^6/6! + 1420103*x^7/7! + 36240369*x^8/8! + 1060638241*x^9/9! + ...

%e such that

%e A(x) = cos(x) + sin(x)*A(x)^2.

%e RELATED SERIES.

%e log(A(x)) = x + 2*x^2/2! + 16*x^3/3! + 160*x^4/4! + 2240*x^5/5! + 39392*x^6/6! + 841216*x^7/7! + 21130240*x^8/8! + 610734080*x^9/9! + ...

%e A(x)^2 = 1 + 2*x + 8*x^2/2! + 64*x^3/3! + 736*x^4/4! + 11072*x^5/5! + 206528*x^6/6! + 4607104*x^7/7! + 119766016*x^8/8! + ...

%o (PARI) {a(n) = my(A = 2*cos(x +x^2*O(x^n)) / (1 + sqrt(1 - 2*sin(2*x +x^2*O(x^n)))) ); n!*polcoeff(A, n)}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A318007.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 21 2018

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Last modified July 1 04:13 EDT 2022. Contains 354951 sequences. (Running on oeis4.)