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%I #8 Nov 19 2021 18:13:29
%S 1,0,2,-3,28,-120,1122,-8127,88096,-885216,11291624,-143432883,
%T 2131731944,-32515910232,555050034224,-9845456006487,190381188822016,
%U -3842126730651264,83143449079579584,-1878918839085535971,45029979676319086976
%N E.g.f (1+arcsin(x))^arcsin(x)
%F a(n)=sum(m=1..n, sum(j=0..(n-m)/2, ((n-2*j)!*stirling1(n-m-2*j,m)*sum(k=0..2*j, (-1)^((3*k)/2)*binomial((n-2)/2,(2*j-k)/2)*sum(i=0..k,(2^i*stirling1(n-2*j+i,n-2*j)*binomial(n-2*j+k-1,n-2*j+i-1))/(n-2*j+i)!)))/(n-m-2*j)!)), n>0, a(0)=1.
%t With[{nn=20},CoefficientList[Series[(1+ArcSin[x])^ArcSin[x],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Nov 19 2021 *)
%o (Maxima)
%o a(n):=sum(sum(((n-2*j)!*stirling1(n-m-2*j,m)*sum((-1)^((3*k)/2)*binomial((n-2)/2,(2*j-k)/2)*sum((2^i*stirling1(n-2*j+i,n-2*j)*binomial(n-2*j+k-1,n-2*j+i-1))/(n-2*j+i)!,i,0,k),k,0,2*j))/(n-m-2*j)!,j,0,(n-m)/2),m,1,n);
%K sign
%O 0,3
%A _Vladimir Kruchinin_, Jun 03 2011