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Expansion of e.g.f. cos(sin(x)*exp(x)).
1

%I #28 Sep 08 2022 08:44:37

%S 1,0,-1,-6,-19,-20,203,1862,9305,20472,-159849,-2441230,-17558715,

%T -60043100,365766243,8445023358,80287239857,383311153776,

%U -1756145007825,-61596647223446,-735340088843107,-4522824431862308,15016682566162427

%N Expansion of e.g.f. cos(sin(x)*exp(x)).

%H G. C. Greubel, <a href="/A009048/b009048.txt">Table of n, a(n) for n = 0..250</a>

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

%p seq(coeff(series(factorial(n)*cos(sin(x)*exp(x)), x,n+1),x,n),n=0..25); # _Muniru A Asiru_, Jul 24 2018

%t With[{nn=30},CoefficientList[Series[Cos[Sin[x]Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jul 10 2015 *)

%o (Maxima)

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

%o (PARI) x='x+O('x^30); Vec(serlaplace(cos(sin(x)*exp(x)))) \\ _G. C. Greubel_, Jul 23 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Cos(Sin(x)*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Jul 23 2018

%K sign,easy

%O 0,4

%A _R. H. Hardin_

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