A331616 - OEIS

%I #13 Jan 26 2020 17:31:02

%S 1,1,3,12,61,380,2783,23240,217817,2267472,25924827,322257408,

%T 4325450325,62374428480,961296291447,15754664717184,273537984529713,

%U 5016337928401152,96871316157146163,1964030207217042432,41706446669511523821,925774982414999202816

%N E.g.f.: exp(1 / (1 - arcsinh(x)) - 1).

%C a(257) is negative. - _Vaclav Kotesovec_, Jan 26 2020

%H Vaclav Kotesovec, <a href="/A331616/b331616.txt">Table of n, a(n) for n = 0..400</a>

%H Vaclav Kotesovec, <a href="/A331616/a331616.jpg">Graph - the asymptotic ratio</a>

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A296675(k) * a(n-k).

%F a(n) ~ 8*(-4*Pi*cos(Pi*(n - 4/(4 + Pi^2))/2) - (Pi^2 - 4)*sin(Pi*(n - 4/(4 + Pi^2))/2)) * n^(n-1) / ((4 + Pi^2)^2 * exp(n + 1 - 4/(4 + Pi^2))). - _Vaclav Kotesovec_, Jan 26 2020

%t nmax = 21; CoefficientList[Series[Exp[1/(1 - ArcSinh[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!

%t A296675[0] = 1; A296675[n_] := A296675[n] = Sum[Binomial[n, k] If[OddQ[k], (-1)^Boole[IntegerQ[(k + 1)/4]] ((k - 2)!!)^2, 0] A296675[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A296675[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

%o (PARI) seq(n)={Vec(serlaplace(exp(1/(1 - asinh(x + O(x*x^n))) - 1)))} \\ _Andrew Howroyd_, Jan 22 2020

%Y Cf. A079484, A296435, A296675, A331607, A331608, A331615, A331617, A331618.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Jan 22 2020