A143139 - OEIS

%I #16 Feb 09 2018 03:10:07

%S 1,3,25,351,6901,174483,5392465,196967991,8301682141,396555037803,

%T 21171512707225,1249311005445231,80742309245690821,

%U 5672134436846492163,430345858647623635105,35069095795843414698471,3054896437732455928549741,283283784773408059496473563

%N E.g.f.: A(x) = exp(x + A(x)^2) - 1.

%F E.g.f.: A(x) = Series_Reversion( log(1+x) - x^2 ).

%F E.g.f. derivative: A'(x) = (1 + A(x))/(1 - 2*A(x) - 2*A(x)^2 ).

%F a(n) = sum(k=0..n-1, (n+k-1)!*sum(j=0..k, (-1)^(j)/(k-j)!*sum(l=0..min(j,(n+j-1)/2), ((-1)^l*stirling1(n-2*l+j-1,j-l))/(l!*(n-2*l+j-1)!)))). - _Vladimir Kruchinin_, Feb 17 2012

%F a(n) ~ sqrt(1+1/sqrt(3)) * 2^(n-3/2) * n^(n-1) / (exp(n) * (sqrt(3)-2-2*log(sqrt(3)-1))^(n-1/2)). - _Vaclav Kotesovec_, Dec 28 2013

%e A(x) = x + 3*x^2/2! + 25*x^3/3! + 351*x^4/4! + 6901*x^5/5! + ...

%e where A(log(1+x) - x^2) = x.

%e Log(1 + A(x)) = x + A(x)^2 = G(x) = g.f. of A143138:

%e G(x) = x + 2*x^2/2! + 18*x^3/3! + 254*x^4/4! + 5010*x^5/5! + ...

%e A(x)^2 = 2*x^2/2! + 18*x^3/3! + 254*x^4/4! + 5010*x^5/5! + ...

%t Rest[CoefficientList[InverseSeries[Series[Log[1+x]-x^2, {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Dec 28 2013 *)

%o (PARI) {a(n)=local(A=x+O(x^n));for(i=0,n,A=exp(x+A^2)-1);n!*polcoeff(A,n)}

%o (PARI) {a(n)=n!*polcoeff(exp(serreverse(x-(exp(x+x*O(x^n))-1)^2))-1,n)}

%o (Maxima)

%o a(n):=sum((n+k-1)!*sum((-1)^(j)/(k-j)!*sum(((-1)^l*stirling1(n-2*l+j-1,j-l))/(l!*(n-2*l+j-1)!),l,0,min(j,(n+j-1)/2)),j,0,k),k,0,n-1); /* _Vladimir Kruchinin_, Feb 17 2012 */

%Y Cf. A143138.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 27 2008