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%I #30 Oct 30 2022 09:00:32
%S 1,1,1,4,13,76,421,3256,25369,245008,2449801,28441216,346065061,
%T 4700478784,67243537453,1047088053376,17192488230961,302112622479616,
%U 5593309059948049,109527844826856448,2255588021494237501
%N Expansion of e.g.f. exp(tan(arcsin(x))).
%H Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F From _Vladimir Kruchinin_, Feb 17 2011: (Start)
%F a(n) = n!*Sum_{k=1..n} A111959(n-1,k-1)*2^(k-n)/k!.
%F a(n) = n!*Sum_{k=1..n} (1+(-1)^(n-k))*C((n-2)/2,(n-k)/2)/(2*k!), n>0.
%F E.g.f.: exp(x/sqrt(1-x^2)). (End)
%F E.g.f.: S(x) = exp(x/sqrt(1-x^2)) = 1 + 2*(x/sqrt(1-x^2))/(G(0) - x/sqrt(1-x^2)), G(k) = 8*k + 2 + (x^2)/((1-x^2)*(8*k+6) + x^2/G(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Dec 16 2011
%F a(n) = (3*n^2 - 12*n + 13)*a(n-2) - 3*(n-4)*(n-3)^2*(n-2)*a(n-4) + (n-6)*(n-5)*(n-4)^2*(n-3)*(n-2)*a(n-6). - _Vaclav Kotesovec_, Nov 08 2013
%F a(n) ~ n^(n-1/3) * exp(3/2*n^(1/3)-n) / sqrt(3) * (1 - 19/(36*n^(1/3)) + 553/(2592*n^(2/3))). - _Vaclav Kotesovec_, Nov 08 2013
%e exp(tan(arcsin(x))) = 1+x+1/2!*x^2+4/3!*x^3+13/4!*x^4+76/5!*x^5...
%p A012150 := proc(n) if n = 0 then 1; else add( (1+(-1)^(n-k)) *binomial((n-2)/2,(n-k)/2)/(2*k!), k=1..n) ; %*n! ; end if; end proc: # _R. J. Mathar_, Mar 20 2011
%t Range[0, 20]! CoefficientList[Series[Exp[Tan[ArcSin[x]]], {x, 0, 20}], x] (* Or *)
%t f[n_] := n! Sum[(1 + (-1)^(n - k)) Binomial[(n - 2)/2, (n - k)/2]/2/k!, {k, n}]; f[0] = 1; Array[f, 21, 0] (* _Robert G. Wilson v_, Feb 19 2011 *)
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(tan(asin(x))))) \\ _Michel Marcus_, Oct 30 2022
%Y Cf. A111959.
%K nonn
%O 0,4
%A Patrick Demichel (patrick.demichel(AT)hp.com)
%E Name edited by _Michel Marcus_, Oct 30 2022
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