OFFSET
0,3
LINKS
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 784
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 + x*log(1-x)) = x. - Paul D. Hanna, Aug 28 2008
E.g.f. A(x) satisfies [from Paul D. Hanna, Jul 15 2012]:
(1) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*(-log(1-x))^n/n!.
(2) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*(-log(1-x))^n/n! ).
a(n) = Sum_{k=0..n-1} k!*(-1)^(n+k-1)*Stirling1(n-1,k)*binomial(n+k-1,n-1). - Vladimir Kruchinin, Feb 01 2012
Lim_{n->infinity} a(n)^(1/n)/n = (1+r)*(2+r)/exp(1) = 1.84542896220833..., where r = 0.794862961852611133... is the root of the equation (1+r)*(r+LambertW(-1,-r*exp(-r))) = -r. - Vaclav Kotesovec, Sep 24 2013
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 188*x^4/4! + 3300*x^5/5! + ...
where A(x) = x - A(x)*log(1-A(x)).
The e.g.f. satisfies [from Paul D. Hanna, Jul 15 2012]:
(1) A(x) = x - x*log(1-x) + d/dx x^2*log(1-x)^2/2! - d^2/dx^2 x^3*log(1-x)^3/3! + d^3/dx^3 x^4*log(1-x)^4/4! + ...
(2) log(A(x)/x) = -log(1-x) + d/dx x*log(1-x)^2/2! - d^2/dx^2 x^2*log(1-x)^3/3! + d^3/dx^3 x^3*log(1-x)^4/4! + ...
MAPLE
spec := [S, {C=Sequence(B), B=Cycle(S), S=Prod(C, Z)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
Flatten[{0, Table[Sum[k!*Abs[StirlingS1[n-1, k]]*Binomial[n+k-1, n-1], {k, 0, n-1}], {n, 1, 20}]}] (* Vaclav Kotesovec, Sep 24 2013 *)
(1+r)*(2+r)/E/.FindRoot[(1+r)*(r+LambertW[-1, -E^(-r)*r]) == -r, {r, 1/2}, WorkingPrecision->50] (* program for numerical value of the limit n->infinity a(n)^(1/n)/n, Vaclav Kotesovec, Sep 24 2013 *)
PROG
(PARI) a(n)=n!*polcoeff(serreverse(x+x*log(1-x +x*O(x^n))), n) \\ Paul D. Hanna, Aug 28 2008
(Maxima) a(n):=(sum(k!*(-1)^(n+k-1)*stirling1(n-1, k)*binomial(n+k-1, n-1), k, 0, n-1)); /* Vladimir Kruchinin, Feb 01 2012 */
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*(-log(1-x+x*O(x^n)))^m/m!)); n!*polcoeff(A, n)} \\ Paul D. Hanna, Jul 15 2012
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*(-log(1-x+x*O(x^n)))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)} \\ Paul D. Hanna, Jul 15 2012
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
STATUS
approved