|
%I #22 Apr 07 2019 19:04:29
%S 1,-4,40,-656,15008,-440896,15821440,-670763264,32806349312,
%T -1818238034944,112618994575360,-7709249275990016,577965256979161088,
%U -47096523207273496576,4144654003816138178560,-391753493233853247586304
%N E.g.f. A(x) = Sum_{n>0} a(n)*x^n/n! is the inverse function to exp(2*x)-x-1.
%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1211.3244">The method for obtaining expressions for coefficients of reverse generating functions</a>, arXiv:1211.3244 [math.CO], 2012.
%F a(n) = sum(k=1..n-1, (n+k-1)!*sum(j=1,k, (-1)^j/(k-j)!*sum(i=0..j, (-1)^i* 2^(n-i+j-1)*stirling2(n-i+j-1,j-i)/((n-i+j-1)!*i!)))), n>1, a(1)=1.
%F a(n) ~ (-1)^(n+1) * 2^(n-1) * n^(n-1) / (exp(n) * (1-log(2))^(n-1/2)). - _Vaclav Kotesovec_, Jan 26 2014
%F a(n) = 2*(1-n)*a(n-1) - Sum_{j=1..n-1} binomial(n,j)*a(j)*a(n-j) for n>1, a(1)=1. - _Peter Luschny_, May 24 2017
%p A205671_list := proc(len) local A, n; A[1] := 1; for n from 2 to len do
%p A[n] := 2*(1-n)*A[n-1] - add(binomial(n,j)*A[j]*A[n-j], j=1..n-1) od:
%p convert(A,list) end: A205671_list(16); # _Peter Luschny_, May 24 2017
%t Rest[CoefficientList[InverseSeries[Series[-1 + E^(2*x) - x,{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 26 2014 *)
%o (Maxima) a(n):=if n=1 then 1 else (sum((n+k-1)!*sum((-1)^j/(k-j)!*sum((-1)^i*2^(n-i+j-1)*stirling2(n-i+j-1,j-i)/((n-i+j-1)!*i!),i,0,j),j,1,k),k,1,n-1));
%o (PARI)
%o x='x+O('x^66); /* that many terms */
%o v=Vec(serlaplace(serreverse(exp(2*x)-x-1)))
%K sign
%O 1,2
%A _Vladimir Kruchinin_, Jan 30 2012
|
