%I #14 Mar 27 2019 10:01:15

%S 0,1,2,5,12,49,190,1301,7224,69441,495898,6095429,53005700,792143793,

%T 8110146070,142633278997,1679413757168,33964965659649,451969255722162,

%U 10331348137881349,153288815339260796,3907452790559751857,63949589015139119598,1798373345567005989781,32179694275204166066728

%N Expansion of e.g.f. arctanh(x)*exp(x).

%F E.g.f.: log((1 + x)/(1 - x))*exp(x)/2.

%F From _Emanuele Munarini_, Dec 16 2017: (Start)

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

%F a(n+3) - a(n+2) - (n+1)*(n+2)*a(n+1) + (n+1)*(n+2)*a(n) = 1.

%F a(n+4) - 2*a(n+3) - (n^2+5*n+5)*a(n+2) + 2*(n+2)^2*a(n+1) - (n+1)*(n+2)*a(n) = 0.

%F (End)

%F a(n) ~ (n-1)! * (exp(1) - (-1)^n * exp(-1))/2. - _Vaclav Kotesovec_, Dec 16 2017

%e E.g.f.: A(x) = x/1! + 2*x^2/2! + 5*x^3/3! + 12*x^4/4! + 49*x^5/5! + ...

%p a:=series(arctanh(x)*exp(x),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # _Paolo P. Lava_, Mar 27 2019

%t nmax = 24; Range[0, nmax]! CoefficientList[Series[ArcTanh[x] Exp[x], {x, 0, nmax}], x]

%t nmax = 24; Range[0, nmax]! CoefficientList[Series[Log[(1 + x)/(1 - x)] Exp[x]/2, {x, 0, nmax}], x]

%t nmax = 24; Range[0, nmax]! CoefficientList[Series[Sum[x^(2 k + 1)/(2 k + 1), {k, 0, Infinity}] Exp[x], {x, 0, nmax}], x]

%t Table[Sum[Binomial[n+1,2k+1](n-2k)/(n+1) (2 k)!, {k,0,n/2}],{n,0,12}] (* _Emanuele Munarini_, Dec 16 2017 *)

%o (Maxima) makelist(sum(binomial(n+1,2*k+1)*(n-2*k)/(n+1)*(2*k)!,k,0,floor(n/2)),n,0,12); /* _Emanuele Munarini_, Dec 16 2017 */

%o (PARI) first(n) = x='x+O('x^n); Vec(serlaplace(atanh(x)*exp(x)), -n) \\ _Iain Fox_, Dec 16 2017

%Y Cf. A002104, A002741, A009739, A009832, A010050, A012709, A087208 (first differences), A279927.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 24 2017