A255432 Expansion of e.g.f. sqrt(1-x^2) * exp(x*(1+arctanh(x))).

1, 1, 2, 4, 12, 36, 160, 680, 4368, 24976, 219616, 1599104, 18145600, 160805568, 2245960704, 23437238656, 389201754368, 4669754525952, 89893314615808, 1218303508421632, 26673872401980416, 402798557886395392, 9883555312758398976

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = a(n-1) + Sum_{k=1..floor(n/2)} (n-1)!/(n-2*k)! * (1/(2*k-1)) * a(n-2*k), a(0)=1.

a(n) ~ (n-2)! * (exp(1) + (-1)^n*exp(-1)). - Vaclav Kotesovec, Feb 24 2015

E.g.f. A(x) satisfies A'(x)/A(x) = 1 + arctanh(x). - Seiichi Manyama, Mar 31 2026

Mathematica

CoefficientList[Series[Sqrt[1-x^2]*E^(x*(1+Log[(1+x)/(1-x)]/2)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 24 2015 *)

Prog

(Maxima)
a(n):=if n=0 then 1 else a(n-1)+sum((n-1)!/(n-2*k)!*(1/(2*k-1))*a(n-2*k), k, 1, n/2);
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sqrt(1-x^2)*exp(x*(1+atanh(x)))))

Crossrefs

Cf. A211393, A394727.

Sequence in context: A276230 A393804 A003701 * A275539 A390493 A356062

Adjacent sequences: A255429 A255430 A255431 * A255433 A255434 A255435

Keyword

nonn

Author

Vladimir Kruchinin, Feb 23 2015

Status

approved