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Expansion of e.g.f. log(1+sin(x)/exp(x)).
0

%I #19 Sep 03 2022 23:17:11

%S 0,1,-3,10,-50,340,-2888,29440,-350160,4759760,-72787488,1236761920,

%T -23115758720,471323145280,-10410977045888,247656022739200,

%U -6312036805140480,171600628707334400,-4956751714926617088

%N Expansion of e.g.f. log(1+sin(x)/exp(x)).

%F E.g.f.: log(1+sin(x)/exp(x)).

%F a(n) = 2*Sum_(k=1..n, Sum_(j=0..(n-k)/2, C(n,n-k-2*j)*(k^(n-k-2*j) *Sum_(i=0..k/2, (2*i-k)^(k+2*j)*C(k,i)*(-1)^(k+j-i))))/(2^k*k)). - _Vladimir Kruchinin_, Jun 13 2011

%F a(n) ~ (-1)^(n+1) * (n-1)! / r^n, where r = 0.588532743981861... is the real root of the equation sin(r) = exp(-r). - _Vaclav Kotesovec_, Oct 25 2013

%t CoefficientList[Series[Log[1+Sin[x]/Exp[x]], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 25 2013 *)

%o (Maxima)

%o a(n):=2*sum(sum(binomial(n,n-k-2*j)*(k^(n-k-2*j)*sum((2*i-k)^(k+2*j) *binomial(k,i)*(-1)^(k+j-i),i,0,k/2)),j,0,(n-k)/2)/(2^k*k),k,1,n); [_Vladimir Kruchinin_, Jun 13 2011]

%K sign,easy

%O 0,3

%A _R. H. Hardin_

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