OFFSET
0,3
FORMULA
E.g.f.: log((1 + x)/(1 - x))*exp(x)/2.
From Emanuele Munarini, Dec 16 2017: (Start)
a(n) = Sum_{k=0..n/2} binomial(n+1,2*k+1)*((n-2*k)/(n+1))*(2*k)!.
a(n+3) - a(n+2) - (n+1)*(n+2)*a(n+1) + (n+1)*(n+2)*a(n) = 1.
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.
(End)
a(n) ~ (n-1)! * (exp(1) - (-1)^n * exp(-1))/2. - Vaclav Kotesovec, Dec 16 2017
EXAMPLE
E.g.f.: A(x) = x/1! + 2*x^2/2! + 5*x^3/3! + 12*x^4/4! + 49*x^5/5! + ...
MAPLE
a:=series(arctanh(x)*exp(x), x=0, 25): seq(n!*coeff(a, x, n), n=0..24); # Paolo P. Lava, Mar 27 2019
MATHEMATICA
nmax = 24; Range[0, nmax]! CoefficientList[Series[ArcTanh[x] Exp[x], {x, 0, nmax}], x]
nmax = 24; Range[0, nmax]! CoefficientList[Series[Log[(1 + x)/(1 - x)] Exp[x]/2, {x, 0, nmax}], x]
nmax = 24; Range[0, nmax]! CoefficientList[Series[Sum[x^(2 k + 1)/(2 k + 1), {k, 0, Infinity}] Exp[x], {x, 0, nmax}], x]
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 *)
PROG
(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 */
(PARI) first(n) = x='x+O('x^n); Vec(serlaplace(atanh(x)*exp(x)), -n) \\ Iain Fox, Dec 16 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 24 2017
STATUS
approved