A205671 E.g.f. A(x) = Sum_{n>0} a(n)*x^n/n! is the inverse function to exp(2*x)-x-1.

1, -4, 40, -656, 15008, -440896, 15821440, -670763264, 32806349312, -1818238034944, 112618994575360, -7709249275990016, 577965256979161088, -47096523207273496576, 4144654003816138178560, -391753493233853247586304

Offset

1,2

Links

Table of n, a(n) for n=1..16.

Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.

Formula

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.

a(n) ~ (-1)^(n+1) * 2^(n-1) * n^(n-1) / (exp(n) * (1-log(2))^(n-1/2)). - Vaclav Kotesovec, Jan 26 2014

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

Maple

A205671_list := proc(len) local A, n; A[1] := 1; for n from 2 to len do
A[n] := 2*(1-n)*A[n-1] - add(binomial(n, j)*A[j]*A[n-j], j=1..n-1) od:
convert(A, list) end: A205671_list(16); # Peter Luschny, May 24 2017

Mathematica

Rest[CoefficientList[InverseSeries[Series[-1 + E^(2*x) - x, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 26 2014 *)

Prog

(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));
(PARI)
x='x+O('x^66); /* that many terms */
v=Vec(serlaplace(serreverse(exp(2*x)-x-1)))

Crossrefs

Sequence in context: A049308 A227055 A343446 * A234294 A181088 A005431

Adjacent sequences: A205668 A205669 A205670 * A205672 A205673 A205674

Keyword

sign

Author

Vladimir Kruchinin, Jan 30 2012

Status

approved