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

1,2

Links

Table of n, a(n) for n=1..18.

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

Example

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! + ...

Mathematica

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

Prog

(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)}
(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 */

Crossrefs

Cf. A143138.

Sequence in context: A093360 A161629 A129506 * A231637 A295765 A012481

Adjacent sequences: A143136 A143137 A143138 * A143140 A143141 A143142

Keyword

nonn

Author

Paul D. Hanna, Jul 27 2008

Status

approved