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%I #32 May 22 2017 12:59:41
%S 1,1,1,0,-3,-4,21,80,-263,-2224,4841,88960,-99723,-4942144,-199939,
%T 366928640,501445617,-35219691264,-101818966319,4251941253120,
%U 19731909099757,-631113275843584,-4192563651606299,113005305852006400,1009030667701246697
%N E.g.f.: exp(arcsin(tanh(x))).
%F a(n) = (-i)^n * Z(n,i), where i = sqrt(-1) and Z(n,x) denotes the n-th zigzag polynomial as described in A147309. Alternative form of the egf: {sec(i*x) - tan(i*x)}^i. - _Peter Bala_, Jan 26 2011
%F a(n)=sum(m=1..n, sum(r=m..n, (sum(k=r..n, (-1)^((3*k)/2)*(sum(i=0..k, (2^i*stirling1(m+i,m)* binomial(m+k-1,m+i-1))/(m+i)!))*binomial((r-2)/2,(r-m-k)/2)))*((-1)^(r-m)+1)*sum(k=0..r-m, binomial(k-1,r-1)*k!*2^(n-k)*stirling2(n,k)*(-1)^(r+k))))/2, n>0, a(0)=1. - _Vladimir Kruchinin_, Jun 09 2011
%F a(n) = sum(i=0..n-1, binomial(n-1,i)*euler(i)*a(n-i-1)), a(0)=1. - _Vladimir Kruchinin_, Feb 26 2015
%e exp(arcsin(tanh(x))) = 1 + x + 1/2!*x^2 - 3/4!*x^4 - 4/5!*x^5 + 21/6!*x^6 ...
%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, i]*EulerE[i]*a[n-i-1], {i, 0, n-1}]; Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, May 22 2017, after _Vladimir Kruchinin_ *)
%o (Maxima)
%o a(n):=sum(sum((sum((-1)^((3*k)/2)*(sum((2^i*stirling1(m+i,m)* binomial(m+k-1,m+i-1))/(m+i)!,i,0,k))*binomial((r-2)/2,(r-m-k)/2),k,0,r-m))*((-1)^(r-m)+1)*sum(binomial(k-1,r-1)*k!*2^(n-k)*stirling2(n,k)*(-1)^(r+k),k,r,n),r,m,n),m,1,n)/2; /* _Vladimir Kruchinin_, Jun 09 2011 */
%o (Maxima)
%o a(n):=if n=0 then 1 else sum(binomial(n-1,i)*euler(i)*a(n-i-1),i,0,n-1); /* _Vladimir Kruchinin_, Feb 26 2015 */
%o (PARI) x='x+O('x^66); Vec(serlaplace(exp(asin(tanh(x))))) \\ _Joerg Arndt_, Feb 26 2015
%K sign
%O 0,5
%A Patrick Demichel (patrick.demichel(AT)hp.com)
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