%I #40 Apr 05 2016 07:32:23
%S 1,1,4,49,1303,63513,5044601,598488981,99463522845,22073876512113,
%T 6308016788410641,2256148067062888845,987271755178677563541,
%U 518851042593331302909225,322466959923499314299129625,233940983258826325064978994501,195913817323275641425583595611805
%N Expansion of exp(x*arcsin(x)) = sum_{n>=0} a(n)*x^(2n)/product(k, 0<k<2n).
%C Expansion of exp(sqrt(x)*arcsin(sqrt(x))) = sum_{n>=0} a(n)*x^(n)/product(k, 0<k<2n). - _Daniel Forgues_, Mar 05 2013
%H Alois P. Heinz, <a href="/A191301/b191301.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = (2*n-1)!*sum(m=1..n, sum(k=0..n, (-1)^k*C((2*n-m-2)/2,n-m-k)* sum(i=0..2*k, (2^i*stirling1(i+m,m)*C(2*k+m-1,i+m-1))/(i+m)!))), n>0, a(0)=1.
%F a(n) ~ exp(Pi/2) * 2^(2*n-1) * n^(2*n-2) / exp(2*n). - _Vaclav Kotesovec_, Apr 05 2016
%e exp(x*arcsin(x)) =
%e 1 + 1/1!*x^2 + 4/3!*x^4 + 49/5!*x^6 + 1303/7!*x^8 + 63513/9!*x^10 + ...
%e exp(sqrt(x)*arcsin(sqrt(x))) =
%e 1 + 1/1!*x^1 + 4/3!*x^2 + 49/5!*x^3 + 1303/7!*x^4 + 63513/9!*x^5 + ...
%t a[n_] := (2*n-1)!* Sum[ Sum[ (Sum[ (2^i*StirlingS1[i+m, m]* Binomial[2*k+m-1, i+m-1])/(i+m)!, {i, 0, 2*k}])*(-1)^k*Binomial[(2*n-m-2)/2, n-m-k], {k, 0, n}], {m, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Mar 04 2013, translated from Maxima *)
%o (Maxima)
%o a(n):=(2*n-1)!*sum(sum((sum((2^i*stirling1(i+m,m)*binomial(2*k+m-1,i+m-1))/ (i+m)!,i,0,2*k))*(-1)^k*binomial((2*n-m-2)/2,n-m-k),k,0,n),m,1,n);
%Y Cf. A042972.
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, May 30 2011