cp's OEIS Frontend

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

%I A101927 #35 Jan 18 2025 01:59:00
%S A101927 1,-2,20,-520,26000,-2132000,260104000,-44217680000,9993195680000,
%T A101927 -2898026747200000,1049085682486400000,-463695871658988800000,
%U A101927 245758811979264064000000,-153845016299019304064000000,112306861898284091966720000000,-94562377718355205435978240000000,90969007365057707629411066880000000
%N A101927 E.g.f. of sin(arcsinh(x)) (odd powers only).
%C A101927 Absolute values are expansion of sinh(arcsin(x)).
%H A101927 Muniru A Asiru, <a href="/A101927/b101927.txt">Table of n, a(n) for n = 1..100</a>
%F A101927 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);
%F A101927 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
%F A101927 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
%F A101927 a(n) ~ (-1)^(n+1) * cosh(Pi/2) * 2^(2*n-1) * n^(2*n-2) / exp(2*n). - _Vaclav Kotesovec_, Oct 23 2013
%F A101927 |a(n+2)| = Product_{k=0..n} ((2k+1)^2+1). - _Andrew Slattery_, Jul 03 2022
%e A101927 sin(arcsinh(x)) = x - 2x^3/3! + 20x^5/5! - 520x^7/7! + 26000x^9/9! - ...
%p A101927 seq(coeff(series(factorial(n)*sin(arcsinh(x)), x,n+1),x,n),n=1..30,2); # _Muniru A Asiru_, Jul 22 2018
%t A101927 Table[n!*SeriesCoefficient[Sin[ArcSinh[x]],{x,0,n}],{n,1,40,2}] (* _Vaclav Kotesovec_, Oct 23 2013 *)
%Y A101927 Bisection of A006228.
%K A101927 sign
%O A101927 1,2
%A A101927 _Ralf Stephan_, Dec 28 2004
%E A101927 Name corrected by _Andrew Slattery_, Jul 03 2022