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A009737 Expansion of e.g.f. tan(x)*exp(tan(x)).

%I A009737 #38 Apr 16 2025 09:01:29
%S A009737 0,1,2,5,20,81,438,2477,16680,120481,973034,8496245,80252732,
%T A009737 817734321,8859646110,102873611549,1258403748432,16372688411713,
%U A009737 223202277906386,3213260867586149,48295209177888356,761792907575450385,12510350648500199814,214507625428065409805
%N A009737 Expansion of e.g.f. tan(x)*exp(tan(x)).
%H A009737 G. C. Greubel, <a href="/A009737/b009737.txt">Table of n, a(n) for n = 0..475</a>
%F A009737 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 A009737 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 A009737 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 A009737 With[{nn=20},CoefficientList[Series[Tan[x]Exp[Tan[x]],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Oct 30 2011 *)
%o A009737 (Maxima)
%o A009737 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 A009737 (Sage)
%o A009737 [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 A009737 (Magma)
%o A009737 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A009737 [0] cat Coefficients(R!( Laplace( Tan(x)*Exp(Tan(x)) ) )); // _G. C. Greubel_, Mar 09 2021
%Y A009737 Cf. A052852, A059419, A008277.
%K A009737 nonn,easy
%O A009737 0,3
%A A009737 _R. H. Hardin_
%E A009737 Extended and signs tested by _Olivier Gérard_, Mar 15 1997