A331615 - OEIS

%I #7 Jan 22 2020 20:14:10

%S 1,1,3,14,85,640,5703,58760,685353,8925632,128231627,2014061568,

%T 34312150525,630043097216,12400033125647,260357810321664,

%U 5807790344591953,137144754146230272,3417248676737769619,89590823377278496768,2465026658283881339301

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

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A189780(k) * a(n-k).

%t nmax = 20; CoefficientList[Series[Exp[1/(1 - ArcSin[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!

%t A189780[0] = 1; A189780[n_] := A189780[n] = Sum[Binomial[n, k] If[OddQ[k], ((k - 2)!!)^2, 0] A189780[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A189780[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

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

%Y Cf. A006228, A189780, A189815, A331607, A331608, A331616, A331617, A331618.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jan 22 2020