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

0, 1, 2, 5, 20, 81, 438, 2477, 16680, 120481, 973034, 8496245, 80252732, 817734321, 8859646110, 102873611549, 1258403748432, 16372688411713, 223202277906386, 3213260867586149, 48295209177888356, 761792907575450385, 12510350648500199814, 214507625428065409805

Offset: 0

Author

R. H. Hardin

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..475

Crossrefs

Cf. A052852, A059419, A008277.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30);
[0] cat Coefficients(R!( Laplace( Tan(x)*Exp(Tan(x)) ) )); // G. C. Greubel, Mar 09 2021

Maple

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

Mathematica

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

Maxima

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 */

Sage

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

Formula

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

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

Extensions

Extended and signs tested by Olivier Gérard, Mar 15 1997