A088691 E.g.f.: A(x) = f(x*A(x)^2), where f(x) = exp(arctan(x)).

1, 1, 5, 47, 657, 12245, 285805, 8022555, 263276705, 9892965545, 418911700725, 19738761470375, 1024422336336625, 58067265415960125, 3569400983720767325, 236508279434832201875, 16804378746368557826625, 1274542376742001037932625, 102780751359763333970849125

Offset

0,3

Comments

Radius of convergence of A(x): r = exp(-Pi/2) = 0.207879576..., with A(r) = exp(Pi/4) = 2.19328..., where r = limit_{n->oo} a(n)/a(n+1)*(n+1). Radius of convergence is from a general formula based on an heuristic argument.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Vaclav Kotesovec, Asymptotic of implicit functions if Fww = 0

Formula

a(n) = n! * [x^n] (exp(arctan(x)))^(2n+1)/(2n+1).

a(n) ~ Gamma(1/3) * exp(n*(Pi/2-1) + Pi/4) * n^(n-5/6) / (2*6^(1/6)*sqrt(Pi)) * (1 - c/n^(1/3)), where c = 0.4593... - Vaclav Kotesovec, Jan 24 2014

a(n) = i^n * (n!/(2*n+1)) * Sum_{k=0..n} (-1)^k * binomial(-i*(n+1/2), n-k) * binomial(i*(n+1/2), k), where i is the imaginary unit. - Fabian Pereyra, Apr 27 2026

Mathematica

Table[n!*SeriesCoefficient[(Exp[ArcTan[x]])^(2n+1)/(2n+1), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 24 2014 *)

Prog

(PARI) a(n)=n!*polcoeff((exp(atan(x)))^(2*n+1)+x*O(x^n), n, x)/(2*n+1)

Crossrefs

Sequence in context: A396678 A180254 A127696 * A052802 A098799 A270529

Adjacent sequences: A088688 A088689 A088690 * A088692 A088693 A088694

Keyword

nonn

Author

Paul D. Hanna, Oct 06 2003

Status

approved