A333691 Expansion of e.g.f. arctan(1 - sqrt(1 - x^2)) (even powers only).

1, 3, 15, 315, 36855, 4833675, 711485775, 133449190875, 33399969978375, 10845524928112875, 4368604540935009375, 2121018409773134746875, 1222083076784378918484375, 826013017674132244878796875, 647724113841936142199672859375, 583169643524919352829283528046875

Offset

1,2

Links

Table of n, a(n) for n=1..16.

Deping Huang, The bowl sequence

Formula

a(1) = 1, a(n+1) = n*(2*n+1)*a(n) + (1/2^(2*n))*Sum_{k=1..n-1} A(n,k)*a(k),

A(n,k) = 2^(2*k)*(2*n-2*k-2)!*(4*n-4*k-5)*(2*n+1)!/((2*k-1)*(n-k+1)*(n-k-1)).

Example

a(1) = 1.
a(2) = 1*(2*1+1)*a(1) = 3.
a(3) = -15*a(1) + 2*(2*2+1)*a(2) = 15.

Mathematica

m = 32; (Range[0, m]! * CoefficientList[Series[ArcTan[1 - Sqrt[1 - x^2]], {x, 0, m}], x])[[3 ;; -1 ;; 2]] (* Amiram Eldar, Apr 04 2020 *)

Prog

(PARI) a(n)={(2*n)!*polcoef(atan(1 - sqrt(1 - x^2 + O(x*x^(2*n)))), 2*n)} \\ Andrew Howroyd, Apr 04 2020

Crossrefs

Sequence in context: A070234 A036279 A156769 * A029758 A103031 A338183

Adjacent sequences: A333688 A333689 A333690 * A333692 A333693 A333694

Keyword

nonn

Author

Deping Huang, Apr 02 2020

Status

approved