A012103 cos(arcsin(sinh(x))) = 1-1/2!*x^2-7/4!*x^4-121/6!*x^6-5167/8!*x^8...

1, -1, -7, -121, -5167, -410641, -51771127, -9466034761, -2365187249887, -773771636088481, -320901749327353447, -164490143309272987801, -102106049239499080993807, -75475990780239097513548721, -65508396610710331890698682967, -65965490141652783720795824187241

Offset

0,3

Links

Table of n, a(n) for n=0..15.

Peter Bala, The method of Graves for computing inverse functions.

Formula

For n>0, a(n) = Sum_{k=1..n} Sum_{j=0..k-1} Pochhammer(-1/2,k) * binomial(2k,j) * (-1)^j*2^(1+2n-2k)*(j-k)^(2n)/k!. - Benedict W. J. Irwin, May 25 2017

E.g.f.: sqrt(1 - sinh(x)^2). - Peter Bala, Feb 20 2026

Maple

# makes use of Graves' method for computing inverse functions
d := proc (n, x) option remember; if n = 0 then sqrt(1 - x^2) else simplify( sqrt(1 + x^2)*diff(d(n-1, x), x) ) end if end proc:
seq( eval(d(2*n, x), x = 0), n = 0..20 ); # Peter Bala, Feb 20 2026

Mathematica

With[{nn=30}, Take[CoefficientList[Series[Cos[ArcSin[Sinh[x]]], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Sep 07 2015 *)
MaxVal = 13; Join[{1}, Table[ Sum[Pochhammer[-(1/2), k] Binomial[2 k, j] (-1)^(j) 2^(1 + 2 i - 2 k) (j - k)^(2 i)/k!, {k, 1, MaxVal}, {j, 0, k - 1}], {i, 1, MaxVal}]] (* Benedict W. J. Irwin, May 25 2017 *)

Crossrefs

Cf. A235134, A385304.

Sequence in context: A012043 A316730 A211103 * A012086 A074487 A192566

Adjacent sequences: A012100 A012101 A012102 * A012104 A012105 A012106

Keyword

sign,easy

Author

Patrick Demichel

Status

approved