A101927 E.g.f. of sin(arcsinh(x)) (odd powers only).

1, -2, 20, -520, 26000, -2132000, 260104000, -44217680000, 9993195680000, -2898026747200000, 1049085682486400000, -463695871658988800000, 245758811979264064000000, -153845016299019304064000000

Offset

1,2

Comments

Absolute values are expansion of sinh(arcsin(x)).

Links

Muniru A Asiru, Table of n, a(n) for n = 1..100

Formula

E.g.f.: sin(arcsinh(x)) = x*sqrt(1+x^2)*(1 - 5*x^2/(G(0) + 5*x^2))); G(k) = (2*k+2)*(2*k+3) - x^2*(4*k^2+8*k+5) + x^2*(2*k+2)*(2*k+3)*(4*k^2+16*k+17)/G(k+1);

for sinh(arcsin(x)) = x*sqrt(1-x^2)*(1 + 5*x^2/(G(0) - 5*x^2))); G(k) = (2*k+2)*(2*k+3) + x^2*(4*k^2+8*k+5) - x^2*(2*k+2)*(2*k+3)*(4*k^2+16*k+17)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 19 2011

G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 + (4*k^2+4*k+2)/(1-x/(x - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013

a(n) ~ (-1)^(n+1) * cosh(Pi/2) * 2^(2*n-1) * n^(2*n-2) / exp(2*n). - Vaclav Kotesovec, Oct 23 2013

|a(n+2)| = Product_{k=0..n} ((2k+1)^2+1). - Andrew Slattery, Jul 03 2022

Example

sin(arcsinh(x)) = x - 2x^3/3! + 20x^5/5! - 520x^7/7! + 26000x^9/9! - ...

Maple

seq(coeff(series(factorial(n)*sin(arcsinh(x)), x, n+1), x, n), n=1..30, 2); # Muniru A Asiru, Jul 22 2018

Mathematica

Table[n!*SeriesCoefficient[Sin[ArcSinh[x]], {x, 0, n}], {n, 1, 40, 2}] (* Vaclav Kotesovec, Oct 23 2013 *)

Crossrefs

Bisection of A006228.

Sequence in context: A103353 A009344 A009699 * A341269 A157317 A350794

Adjacent sequences: A101924 A101925 A101926 * A101928 A101929 A101930

Keyword

sign

Author

Ralf Stephan, Dec 28 2004

Extensions

Name corrected by Andrew Slattery, Jul 03 2022

Status

approved