A104018 E.g.f. (arcsinh(1/sinh(arcsinh(1) - sqrt(2)*x)) - arcsinh(1))/sqrt(2).

0, 1, 2, 6, 28, 180, 1448, 13944, 156592, 2010000, 29026592, 465749856, 8220541888, 158283827520, 3301678947968, 74168218575744, 1785106271372032, 45828856887701760, 1250094695454351872

Offset

0,3

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..417

Formula

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

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

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

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

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.

a(n) ~ (n-1)! * 2^((n+1)/2) / (log(3+2*sqrt(2)) * (log(1+sqrt(2)))^(n-1)). - Vaclav Kotesovec, Jan 07 2014

Example

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

Mathematica

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 *)
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 *)

Prog

(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))};

Crossrefs

Sequence in context: A136631 A002435 A276911 * A100526 A200560 A303344

Adjacent sequences: A104015 A104016 A104017 * A104019 A104020 A104021

Keyword

nonn

Author

Michael Somos, Feb 28 2005

Status

approved