A291483 - OEIS

%I #6 Mar 27 2019 10:01:06

%S 0,1,2,2,0,4,40,-64,-1344,3984,85408,-356896,-8462080,45908160,

%T 1209040768,-8080805888,-235449260032,1871655631104,59955521585664,

%U -552758145525248,-19339870285225984,202927333558572032,7707208199780517888,-90698934927786770432,-3718489569130941169664,48507735629457304555520

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

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

%e E.g.f.: A(x) = x/1! + 2*x^2/2! + 2*x^3/3! + 4*x^5/5! + 40*x^6/6! - 64*x^7/7! - 1344*x^8/8! + ...

%p a:=series(arcsinh(x)*exp(x),x=0,26): seq(n!*coeff(a,x,n),n=0..25); # _Paolo P. Lava_, Mar 27 2019

%t nmax = 25; Range[0, nmax]! CoefficientList[Series[ArcSinh[x] Exp[x], {x, 0, nmax}], x]

%t nmax = 25; Range[0, nmax]! CoefficientList[Series[Log[x + Sqrt[1 + x^2]] Exp[x], {x, 0, nmax}], x]

%t nmax = 25; Range[0, nmax]! CoefficientList[Series[-Sum[((-1)^k (-1 + x + Sqrt[1 + x^2])^k)/k, {k, 1, Infinity}] Exp[x], {x, 0, nmax}], x]

%Y Cf. A001818, A009545, A012584, A131577, A291482.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Aug 24 2017