A291484 Expansion of e.g.f. arctanh(x)*exp(x).

0, 1, 2, 5, 12, 49, 190, 1301, 7224, 69441, 495898, 6095429, 53005700, 792143793, 8110146070, 142633278997, 1679413757168, 33964965659649, 451969255722162, 10331348137881349, 153288815339260796, 3907452790559751857, 63949589015139119598, 1798373345567005989781, 32179694275204166066728

Offset: 0

Author

Ilya Gutkovskiy, Aug 24 2017

Keywords

• nonn

Examples

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

Crossrefs

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

Programs

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 *)

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

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