A104018 - OEIS

%I #26 Oct 03 2018 03:21:50

%S 0,1,2,6,28,180,1448,13944,156592,2010000,29026592,465749856,

%T 8220541888,158283827520,3301678947968,74168218575744,

%U 1785106271372032,45828856887701760,1250094695454351872

%N E.g.f. (arcsinh(1/sinh(arcsinh(1) - sqrt(2)*x)) - arcsinh(1))/sqrt(2).

%D D. M. Y. Sommerville, The Elements of Non-Euclidean Geometry, Dover Publications, 1958, pp. 235, 243. MR0100246 (20 #6679)

%H G. C. Greubel, <a href="/A104018/b104018.txt">Table of n, a(n) for n = 0..417</a>

%F Series reversion of e.g.f. A(x) is -A(-x).

%F E.g.f. A(x)=y satisfies y' = sinh(arcsinh(1) + sqrt(2)*y).

%F E.g.f.: (arcsinh(1/sinh(arcsinh(1)-sqrt(2)*x)) - arcsinh(1))/sqrt(2).

%F With C=sqrt(2): 1/(cosh(C*x)-C*sinh(C*x)) = 1 + 2x + 6x^2/2! + 28x^3/3! + 180x^4/4! + ... - _Ralf Stephan_, Mar 01 2005

%F G.f.: x/G(0) where G(k) = 1 - 2*x*(2*k+1) - 2*x^2*(k+1)*(k+1)/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jan 11 2013.

%F a(n) ~ (n-1)! * 2^((n+1)/2) / (log(3+2*sqrt(2)) * (log(1+sqrt(2)))^(n-1)). - _Vaclav Kotesovec_, Jan 07 2014

%e E.g.f. = x + x^2 + x^3 + 7/6*x^4 + 3/2*x^5 + 181/90*x^6 + 83/30*x^7 + ...

%t Flatten[{0,CoefficientList[Series[1/(Cosh[Sqrt[2]*x]-Sqrt[2]*Sinh[Sqrt[2]*x]), {x, 0, 20}], x]* Range[0, 20]!}] (* _Vaclav Kotesovec_, Jan 07 2014 *)

%t a[ n_] := With[{m = n - 1}, If[ m < 1, Boole[m == 0], m! SeriesCoefficient[ 1 / Sum[ (-x)^k/k! 2^Quotient[k + 1, 2], {k, 0, m}], {x, 0, m}]]]; (* _Michael Somos_, Oct 03 2018 *)

%o (PARI) {a(n) = if( n<2, n>0, n--; n! * polcoeff( 1 / sum(k=0, n, (-x)^k/k! * 2^((k+1)\2), x * O(x^n)), n))};

%K nonn

%O 0,3

%A _Michael Somos_, Feb 28 2005