OFFSET
0,3
COMMENTS
A simple grammar.
LINKS
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 809
Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
FORMULA
E.g.f. satisfies: A(x) = -log(1 - x/(1-A(x))). [From Encyclopedia of Combinatorial Structures]
a(n) = sum(k=0..n-1, (sum(j=0..k, (sum(i=0..j, (stirling2(i+n-1,j)*C(j,j-i))/ (i+n-1)!))*(-1)^(n+j-1)/(k-j)!))*(n+k-1)!), n>0. - Vladimir Kruchinin, Feb 06 2012
a(n) ~ n^(n-1) * c^n / (sqrt(1+c) * exp(n) * (c-1)^(2*n-1)), where c = LambertW(exp(2)) = 1.5571455989976114... (see A226571). - Vaclav Kotesovec, Jan 08 2014
For n >= 1, a(n) = Sum_{k=0..n-1} Pochhammer(n, k)*|Stirling1(n, k+1)|. - Mélika Tebni, Jun 02 2023
EXAMPLE
E.g.f.: A(x) = x + 3*x^2/2! + 23*x^3/3! + 290*x^4/4! + 5104*x^5/5! +... which satisfies: A(x) = -log(1 - x/(1-A(x))).
MAPLE
spec := [S, {C=Prod(Z, B), S=Cycle(C), B=Sequence(S)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
# second Maple program:
A052842 := proc (n) option remember; `if`(n = 0, 0, add(pochhammer(n, k)*abs(Stirling1(n, k+1)), k = 0..n-1)) end:
seq(A052842(n), n = 0..16); # Mélika Tebni, Jun 02 2023
MATHEMATICA
CoefficientList[InverseSeries[Series[(-1 + E^(-x))*(x-1), {x, 0, 20}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 08 2014 *)
PROG
(PARI) {a(n)=n!*polcoeff(serreverse((1-exp(-x+O(x^(n+2))))*(1-x)), n)} /* Paul D. Hanna, Jun 22 2011 */
(Maxima) a(n):=sum((sum((sum((stirling2(i+n-1, j)*binomial(j, j-i))/(i+n-1)!, i, 0, j))*(-1)^(n+j-1)/(k-j)!, j, 0, k))*(n+k-1)!, k, 0, n-1); /* Vladimir Kruchinin, Feb 06 2012 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
Name from a comment by Paul D. Hanna, Jun 22 2011
STATUS
approved