A320959 The exponential limit of arctanh (odd indices only).

1, 10, 1248, 631440, 852647040, 2462394816000, 13241729554099200, 120563962753538304000, 1733764260005567741952000, 37343395325946891151466496000, 1155311729350231354981936496640000, 49626886713956000390638096497377280000, 2878005957927359237424925417166882734080000

Offset: 0

Author

Peter Luschny, Nov 08 2018

Keywords

• nonn

Comments

See A320956 for definitions and comments.

Examples

			Illustration of the convergence in the sense of A320956:
   [0] 0, 0, 0,  0, 0,    0, 0,      0, 0,         0, ...
   [1] 0, 1, 0,  2, 0,   24, 0,    720, 0,     40320, ... A005359
   [2] 0, 1, 0,  8, 0,  384, 0,  46080, 0,  10321920, ... A067624
   [3] 0, 1, 0, 10, 0,  984, 0, 262800, 0, 132289920, ...
   [4] 0, 1, 0, 10, 0, 1224, 0, 514800, 0, 445576320, ...
   [5] 0, 1, 0, 10, 0, 1248, 0, 615600, 0, 725840640, ...
   [6] 0, 1, 0, 10, 0, 1248, 0, 630720, 0, 832527360, ...
   [7] 0, 1, 0, 10, 0, 1248, 0, 631440, 0, 851155200, ...
   [8] 0, 1, 0, 10, 0, 1248, 0, 631440, 0, 852606720, ...
   [9] 0, 1, 0, 10, 0, 1248, 0, 631440, 0, 852647040, ...
		

Crossrefs

Cf. A320955 (exp), A320962 (log(x+1)), A320956 (sec+tan), A320958 (arcsin), this sequence (arctanh).

Cf. A005359, A067624.

Programs

Maple

# The function ExpLim is defined in A320956.
L := [ExpLim(28, arctanh)]: seq(L[2*n], n=1..13);

Mathematica

m = 13; CoefficientList[ArcTanh[x] + O[x]^(2 m + 1), x]*Range[0, 2 m - 1]!*BellB[Range[0, 2 m - 1]] // DeleteCases[#, 0]& (* Jean-François Alcover, Jul 23 2019 *)

Formula

For n >= 3 and odd, -a(m)*Zeta(m) = g(n), where g denotes the exponential limit of log(Gamma(x + 1)) and m = (n-1)/2.