A296437 Expansion of e.g.f. log(1 + arcsinh(x))*exp(x).

0, 1, 1, 1, 0, 8, -5, -51, -504, 8224, -12865, -296155, -2166736, 73348780, -116217309, -7440979651, -39733320080, 2564082122752, -3056854891489, -544155777899859, -2138400746459448, 251904027415707852, -163714875656114029, -92626483427571793931, -273784346863222483272

Offset

0,6

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Vaclav Kotesovec, Graph - The asymptotic ratio

Formula

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

a(n) ~ n! * 2*sqrt(2/Pi) * (Pi*c - 2*s) / (n^(3/2) * (4 + Pi^2)) * (1 + (c*(-192 + 208*Pi - 96*Pi^2 - 8*Pi^3 - 12*Pi^4 + Pi^5) - 2*s*(80 + 48*Pi - 40*Pi^2 + 24*Pi^3 + Pi^4 + 3*Pi^5)) / (4*(4 + Pi^2)^2 * (c*Pi - 2*s)*n)), where s = sin(1 - Pi*n/2) and c = cos(1 - Pi*n/2). - Vaclav Kotesovec, Dec 21 2017

Example

E.g.f.: A(x) = x/1! + x^2/2! + x^3/3! + 8*x^5/5! - 5*x^6/6! - 51*x^7/7! - 504*x^8/8! + ...

Maple

a:=series(log(1+arcsinh(x))*exp(x), x=0, 25): seq(n!*coeff(a, x, n), n=0..24); # Paolo P. Lava, Mar 27 2019

Mathematica

nmax = 24; CoefficientList[Series[Log[1 + ArcSinh[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Log[1 + Log[x + Sqrt[1 + x^2]]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!

Prog

(PARI) my(ox=O(x^30)); Vecrev(Pol(serlaplace(log(1 + asinh(x + ox)) * exp(x + ox)))) \\ Andrew Howroyd, Dec 12 2017

Crossrefs

Cf. A009334, A009353, A009356, A291483, A296435, A296436.

Sequence in context: A272291 A272543 A272007 * A038283 A317402 A317574

Adjacent sequences: A296434 A296435 A296436 * A296438 A296439 A296440

Keyword

sign

Author

Ilya Gutkovskiy, Dec 12 2017

Status

approved