A133941 E.g.f. satisfies: A(x) = x*(exp(tan(A(x)))).

0, 1, 2, 9, 72, 825, 12192, 220353, 4708480, 116116497, 3245839360, 101415497689, 3502465714176, 132486192976137, 5447446920323072, 241907419042038225, 11538444129924055040, 588321821566662253729, 31932991994214557417472

Offset: 0

Author

Alois P. Heinz, Aug 27 2008

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100

Programs

Maple

A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (exp (tan(A(n-1)))), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..22);

Mathematica

CoefficientList[InverseSeries[Series[x/E^Tan[x],{x,0,20}],x],x] * Range[0,20]! (* Vaclav Kotesovec, Jul 16 2014 *)

Maxima

a(n):=if n<2 then n else sum((n^k*((-1)^(n-k-1)+1)*sum(binomial(j-1,k-1)*j!*2^(n-j-2)*(-1)^((n+k-1)/2+j)*stirling2(n-1,j),j,k,n-1))/k!,k,1,n-1) [Vladimir Kruchinin, May 10 2011]

Formula

a(n) = sum(k=1..n-1, (n^k*((-1)^(n-k-1)+1)*sum(j=k..n-1, C(j-1,k-1)*j!*2^(n-j-2) * (-1)^((n+k-1)/2+j)*stirling2(n-1,j),j,k,n-1))/k!) n>1. a(1)=1. - Vladimir Kruchinin, May 10 2011

a(n) ~ n^(n-1) * s / (exp(n) * r^n * sqrt(1+sin(2*s))), where r = 0.3039707088650079908... and s = 0.6417143708728826584... are roots of the system of equations exp(tan(s))*r = s, s = (cos(s))^2. - Vaclav Kotesovec, Jul 16 2014