A009442 E.g.f. log(1 + x/cos(x)).

0, 1, -1, 5, -18, 109, -720, 5977, -56336, 612729, -7453440, 100954061, -1502172672, 24395453861, -429076910080, 8128143367905, -164961704478720, 3571195811862385, -82142328351817728, 2000535014776893973

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Crossrefs

Cf. A003957.

Programs

Mathematica

CoefficientList[Series[Log[1 + x*Sec[x]], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 24 2015 *)

Maxima

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 */

Formula

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

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

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997