A255406 - OEIS

%I #19 Jun 10 2024 23:35:05

%S 1,18,26460,288149400,11799717930000,1303467640855380000,

%T 318564884489773161240000,150951970515479012453574000000,

%U 126206413988876537942059614180000000,173464405707011357574463836709701000000000,370958141678992170468287850863450040726000000000

%N Expansion of the hyperbolic arc lemniscate tangent function.

%C Gauss's hyperbolic arc lemniscate sine function arcslh(x) is defined by arcslh(x) = Integral_{t = 0..x} 1/sqrt(1 + t^4) dt, for x real. Neuman (2007) introduced the hyperbolic arc lemniscate tangent function arctlh(x), defined by arctlh(x) = arcslh( x/(1 - x^4)^(1/4) ) for |x| < 1.

%H Chao-Ping Chen, <a href="http://jmi.ele-math.com/06-65/Wilker-and-Huygens-type-inequalities-for-the-Lemniscate-functions">Wilker and Huygens type inequalities for the lemniscate functions </a>, J. Math. Inequalities Vol. 6, Number 4 (2012), 673-684.

%H E. Neuman, <a href="http://ttk.pte.hu/mii/html/pannonica/index_elemei/vol_18_1_cont.htm">On Gauss lemniscate functions and lemniscatic mean</a>, Mathematica Pannonica, 18 (2007), no. 1, 77-94.

%F a(n) = (n - 1/4)! *(4*n)!/( (-1/4)! * n! ).

%F a(n) = Product_{k = 1..4*n} k - 0^(k mod 4), where we make the usual convention that 0^0 = 1. Cf. A001818 ( Product_{k = 1..2*n} k - 0^(k mod 2) ) and A158111 ( Product {k = 1..3*n} k - 0^(k mod 3) ).

%F G.f.: arctlh(x) = x + 18*x^5/5! + 26460*x^9/9! + 288149400*x^13/13! + ....

%F d/dx( arctlh(x) ) = 1/(1 - x^4)^(3/4) = 1 + 18*x^4/4! + 26460*x^8/8! + 288149400*x^12/12! + ....

%F a(n) ~ (4*n)! / (n^(1/4) * Gamma(3/4)). - _Vaclav Kotesovec_, Feb 22 2015

%e 1/sqrt(1 + t^4) = 1 - (1/2)*t^4 + (3/8)*t^8 - ....

%e arcslh(x) = Integral_{t = 0..x} 1/sqrt(1 + t^4) dt = x - (1/10)*x^5 + (1/24)*x^9 - ....

%e Hence arctlh(x) = x/(1 - x^4)^(1/4) - (1/10)*x^5/(1 - x^4)^(5/4) + (1/24)*x^9/(1 - x^4)^(9/4) - ... = x + 18*x^5/5! + 26460*x^9/9! + ....

%p a:= n-> mul(k-0^(irem(k, 4)), k=1..4*n): seq(a(n), n=0..11);

%t nmax=15; Table[(CoefficientList[Series[1/(1-x^4)^(3/4),{x,0,4*nmax}],x] * Range[0,4*nmax]!)[[4*n-3]],{n,1,nmax}] (* _Vaclav Kotesovec_, Feb 22 2015*)

%t Table[Pochhammer[3/4, n]*(4*n)!/n!, {n, 0, 10}] (* _Jean-François Alcover_, Mar 05 2015 *)

%o (PARI) a(n) = prod(k = 1, 4*n, k - 0^(k % 4)); \\ _Michel Marcus_, Mar 03 2015

%Y Cf. A001818, A158111.

%K nonn,easy

%O 0,2

%A _Peter Bala_, Feb 22 2015