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%I #20 Apr 01 2017 14:16:37
%S 0,1,-1,5,-18,109,-720,5977,-56336,612729,-7453440,100954061,
%T -1502172672,24395453861,-429076910080,8128143367905,-164961704478720,
%U 3571195811862385,-82142328351817728,2000535014776893973
%N E.g.f. log(1 + x/cos(x)).
%F a(n)=2*n!*sum(m=1..(n-1)/2, ((-1)^(n-1)*sum(j=0..m, binomial((n/2-m+j-1),j)*4^(m-j)*sum(i=0..j, (i-j)^(2*m)*binomial(2*j,i)*(-1)^(m+j-i))))/((n-2*m)*(2*m)!))+(-1)^(n-1)*n!/n. - _Vladimir Kruchinin_, Jun 16 2011
%F a(n) ~ (n-1)! * (-1)^(n+1) / r^n, where r = 0.7390851332151606416553120876738734040134117589... (see A003957) is the root of the equation cos(r) = r. - _Vaclav Kotesovec_, Jan 24 2015
%t CoefficientList[Series[Log[1 + x*Sec[x]], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jan 24 2015 *)
%o (Maxima)
%o a(n):=2*n!*sum(((-1)^(n-1)*sum(binomial((n/2-m+j-1),j)*4^(m-j)*sum((i-j)^(2*m)*binomial(2*j,i)*(-1)^(m+j-i),i,0,j),j,0,m))/((n-2*m)*(2*m)!),m,1,(n-1)/2)+(-1)^(n-1)*n!/n; /* _Vladimir Kruchinin_, Jun 16 2011 */
%Y Cf. A003957.
%K sign,easy
%O 0,4
%A _R. H. Hardin_
%E Extended with signs by _Olivier Gérard_, Mar 15 1997
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