A143139 E.g.f.: A(x) = exp(x + A(x)^2) - 1.
1, 3, 25, 351, 6901, 174483, 5392465, 196967991, 8301682141, 396555037803, 21171512707225, 1249311005445231, 80742309245690821, 5672134436846492163, 430345858647623635105, 35069095795843414698471, 3054896437732455928549741, 283283784773408059496473563
Offset: 1
Keywords
Examples
A(x) = x + 3*x^2/2! + 25*x^3/3! + 351*x^4/4! + 6901*x^5/5! + ... where A(log(1+x) - x^2) = x. Log(1 + A(x)) = x + A(x)^2 = G(x) = g.f. of A143138: G(x) = x + 2*x^2/2! + 18*x^3/3! + 254*x^4/4! + 5010*x^5/5! + ... A(x)^2 = 2*x^2/2! + 18*x^3/3! + 254*x^4/4! + 5010*x^5/5! + ...
Crossrefs
Cf. A143138.
Programs
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Mathematica
Rest[CoefficientList[InverseSeries[Series[Log[1+x]-x^2, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Dec 28 2013 *) -
Maxima
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 */
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PARI
{a(n)=local(A=x+O(x^n));for(i=0,n,A=exp(x+A^2)-1);n!*polcoeff(A,n)} -
PARI
{a(n)=n!*polcoeff(exp(serreverse(x-(exp(x+x*O(x^n))-1)^2))-1,n)}
Formula
E.g.f.: A(x) = Series_Reversion( log(1+x) - x^2 ).
E.g.f. derivative: A'(x) = (1 + A(x))/(1 - 2*A(x) - 2*A(x)^2 ).
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
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