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Expansion of e.g.f. tan(x)*exp(tan(x)).
2

%I #31 Mar 10 2021 01:18:26

%S 0,1,2,5,20,81,438,2477,16680,120481,973034,8496245,80252732,

%T 817734321,8859646110,102873611549,1258403748432,16372688411713,

%U 223202277906386,3213260867586149,48295209177888356

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

%H G. C. Greubel, <a href="/A009737/b009737.txt">Table of n, a(n) for n = 0..475</a>

%F a(n) = Sum_{k=1..n} ( ((1+(-1)^(n-k))/(k-1)!) * Sum_{j=k..n} j! * Stirling2(n,j) * 2^(n-j-1)*(-1)^((n+k)/2+j)*binomial(j-1,k-1) ) ). - _Vladimir Kruchinin_, Apr 19 2011

%F a(n) = D^n(x*exp(x)) evaluated at x = 0, where D is the operator (1+x^2)*d/dx. Cf. A052852. a(n) = Sum_{k=1..n} k*A059419(n,k). - _Peter Bala_, Nov 25 2011

%p m:= 30; S:= series(tan(x)*exp(tan(x)), x, m+1); seq(j!*coeff(S, x, j), j = 0..m); # _G. C. Greubel_, Mar 09 2021

%t With[{nn=20},CoefficientList[Series[Tan[x]Exp[Tan[x]],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Oct 30 2011 *)

%o (Maxima)

%o a(n):=sum((1+(-1)^(n-k))*sum(j!*stirling2(n,j)*2^(n-j-1)*(-1)^((n+k)/2+j)*binomial(j-1,k-1),j,k,n)/(k-1)!,k,1,n); [_Vladimir Kruchinin_, Apr 19 2011]

%o (Sage)

%o [factorial(n)*( tan(x)*exp(tan(x)) ).series(x,n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Mar 09 2021

%o (Magma)

%o R<x>:=PowerSeriesRing(Rationals(), 30);

%o [0] cat Coefficients(R!( Laplace( Tan(x)*Exp(Tan(x)) ) )); // _G. C. Greubel_, Mar 09 2021

%Y Cf. A052852, A059419, A008277.

%K nonn,easy

%O 0,3

%A _R. H. Hardin_

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