A255406 Expansion of the hyperbolic arc lemniscate tangent function.

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

Offset

0,2

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.

Links

Table of n, a(n) for n=0..10.

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.

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

Example

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! + ....

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 *)

Prog

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

Crossrefs

Cf. A001818, A158111.

Sequence in context: A262359 A202155 A296653 * A123401 A201494 A079303

Adjacent sequences: A255403 A255404 A255405 * A255407 A255408 A255409

Keyword

nonn,easy

Author

Peter Bala, Feb 22 2015

Status

approved