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%I #21 Mar 05 2024 11:27:06
%S 1,1,3,10,53,304,2303,18768,185033,1954176,23756667,308077056,
%T 4457821821,68513332224,1150764459063,20443736745984,391167511473681,
%U 7884821722497024,169370797497060339,3818539009013907456,91013260219635394629,2269047587255097753600,59435772666287730632559
%N Expansion of e.g.f. log(1/(1-arcsin(x))).
%H G. C. Greubel, <a href="/A189815/b189815.txt">Table of n, a(n) for n = 1..435</a>
%F a(n) = (n-1)!*Sum_{m=1..(n-1)} ((1+(-1)^(n-m))/2)*Sum_{k=1..(n-m)} (Sum_{j=1..k} binomial(k,j)*2^(1-j)*Sum_{i=0..floor(j/2)} (-1)^((n-m)/2-i-j)*binomial(j,i)*(j-2*i)^(n-m+j)/(n-m+j)!)*binomial(k+n-1,n-1)) + (n-1)!.
%F a(n) ~ (n-1)! / (sin(1))^n. - _Vaclav Kotesovec_, Nov 06 2014
%t Rest[With[{nmax = 50}, CoefficientList[Series[Log[1/(1 - ArcSin[x])], {x, 0, nmax}], x]*Range[0, nmax]!]] (* _G. C. Greubel_, Jan 16 2018 *)
%o (Maxima) a(n):=(n-1)!*sum((1+(-1)^(n-m))/2*sum((sum(binomial(k,j)*2^(1-j)*sum((-1)^((n-m)/2-i-j)*binomial(j,i)*(j-2*i)^(n-m+j)/(n-m+j)!,i,0,floor(j/2)),j,1,k))*binomial(k+n-1,n-1),k,1,n-m),m,1,n-1)+(n-1)!;
%o (PARI) x='x+O('x^30); Vec(serlaplace(log(1/(1-asin(x))))) \\ _G. C. Greubel_, Jan 16 2018
%K nonn
%O 1,3
%A _Vladimir Kruchinin_, May 02 2011
%E Terms a(18) onward added by _G. C. Greubel_, Jan 16 2018
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