A296169 E.g.f. A(x) satisfies: A(x) = 1+x - cos(2*A(x) - x).

1, 1, 6, 59, 810, 14281, 307566, 7825859, 229715130, 7640988961, 284037675966, 11669182625099, 525040651527210, 25676859334384441, 1356133254350401806, 76928506160117877779, 4664746297141400237850, 301102611588796277314321, 20613405033136513233790686, 1491812049486032067219356699, 113798761459974922574012320650

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

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

(1) A(x) = 1+x - cos(2*A(x) - x).

(2) A(x) = x + 2*sin(A(x) - x/2)^2.

(3) A(x) = x/2 + Series_Reversion( 2*x + 2*cos(2*x) - 2 ).

(4) A(x) = x/2 + Series_Reversion( 2*x - 4*sin(x)^2 ).

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

Example

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! + ...
such that A(x) = 1+x - cos(2*A(x) - x).

Mathematica

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 *)

Prog

(PARI) {a(n) = my(A = x/2 + serreverse(2*x - 4*sin(x +x*O(x^n))^2) ); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Sequence in context: A114501 A369513 A256035 * A089153 A075136 A024382

Adjacent sequences: A296166 A296167 A296168 * A296170 A296171 A296172

Keyword

nonn

Author

Paul D. Hanna, Feb 05 2018

Status

approved