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E.g.f.: exp(1 / (1 - sin(x)) - 1).
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%I #9 Jan 26 2020 08:18:45

%S 1,1,3,12,61,372,2639,21280,191833,1908688,20750331,244478784,

%T 3100597333,42088689216,608543191559,9332562964480,151252803045937,

%U 2582250195499264,46306562212010355,870011934425816064,17086276243125287917

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

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A000111(k+1) * a(n-k).

%F a(n) ~ 2^(n + 2/3) * exp(8/(3*Pi^2) - 5/6 + 2^(5/3) * n^(1/3) / Pi^(4/3) + 3 * 2^(1/3) * n^(2/3) / Pi^(2/3) - n) * n^(n - 1/6) / (sqrt(3) * Pi^(n + 1/3)). - _Vaclav Kotesovec_, Jan 26 2020

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

%t A000111[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n + 1) (2^(n + 1) - 1) BernoulliB[n + 1])/(n + 1)]]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A000111[k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

%Y Cf. A000111, A000772, A002017, A331608, A331610, A331611.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jan 22 2020