A255406 Expansion of the hyperbolic arc lemniscate tangent function.

1, 18, 26460, 288149400, 11799717930000, 1303467640855380000, 318564884489773161240000, 150951970515479012453574000000, 126206413988876537942059614180000000, 173464405707011357574463836709701000000000, 370958141678992170468287850863450040726000000000

Offset: 0

Author

Peter Bala, Feb 22 2015

Keywords

• nonn
• easy

Comments

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.

Examples

			1/sqrt(1 + t^4) = 1 - (1/2)*t^4 + (3/8)*t^8 - ....
arcslh(x) = Integral_{t = 0..x} 1/sqrt(1 + t^4) dt = x - (1/10)*x^5 + (1/24)*x^9 - ....
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! + ....
		

Links

• Chao-Ping Chen, Wilker and Huygens type inequalities for the lemniscate functions , J. Math. Inequalities Vol. 6, Number 4 (2012), 673-684.
• E. Neuman, On Gauss lemniscate functions and lemniscatic mean, Mathematica Pannonica, 18 (2007), no. 1, 77-94.

Crossrefs

Cf. A001818, A158111.

Programs

Maple

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

Mathematica

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*)
Table[Pochhammer[3/4, n]*(4*n)!/n!, {n, 0, 10}] (* Jean-François Alcover, Mar 05 2015 *)

PARI

a(n) = prod(k = 1, 4*n, k - 0^(k % 4)); \\ Michel Marcus, Mar 03 2015

Formula

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

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) ).

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

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

a(n) ~ (4*n)! / (n^(1/4) * Gamma(3/4)). - Vaclav Kotesovec, Feb 22 2015