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A009377 E.g.f. log(1 + tan(x)*sin(x)) (even powers only). 1

%I

%S 0,2,-8,182,-6008,408122,-38757908,5438711462,-1008011932208,

%T 244100206825202,-74027819501268908,27620824218436349342,

%U -12405602584546021488008,6609444480661620416243882

%N E.g.f. log(1 + tan(x)*sin(x)) (even powers only).

%F a(n)=sum(k=1..2*n, ((-1)*sum(t=0..n-k, binomial(2*n,2*t+k)*((sum(j=k..2*n-2*t-k, binomial(j-1,k-1)*j!*stirling2(2*n-2*t-k,j)*(-1)^(n+j)*2^(-2*t+2*n-2*k-j+1)))*sum(i=0..k/2, (2*i-k)^(2*t+k)*binomial(k,i)*(-1)^(i)))))/(k)). - _Vladimir Kruchinin_, Jun 30 2011

%F a(n) ~ (2*n)! * (-1)^(n+1) / (n * (log((1 + sqrt(5) + sqrt(2*(1 + sqrt(5)))) / 2))^(2*n)). - _Vaclav Kotesovec_, Jan 24 2015

%t nn = 20; Table[(CoefficientList[Series[Log[1 + Sin[x]*Tan[x]], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* _Vaclav Kotesovec_, Jan 24 2015 *)

%o (Maxima)

%o a(n):=sum(((-1)*sum(binomial(2*n,2*t+k)*((sum(binomial(j-1,k-1)*j!*stirling2(2*n-2*t-k,j)*(-1)^(n+j)*2^(-2*t+2*n-2*k-j+1),j,k,2*n-2*t-k))*sum((2*i-k)^(2*t+k)*binomial(k,i)*(-1)^(i),i,0,k/2)),t,0,n-k))/(k),k,1,2*n); /* _Vladimir Kruchinin_, Jun 30 2011 */

%K sign

%O 0,2

%A _R. H. Hardin_

%E Extended with signs by _Olivier GĂ©rard_, Mar 15 1997

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Last modified November 26 21:07 EST 2021. Contains 349344 sequences. (Running on oeis4.)