A138014 E.g.f. A(x) satisfies exp(A(x)) = x + exp(A(x)^2) where A(0) = 0.

1, 1, 2, 16, 174, 1988, 27124, 453136, 8791980, 191869392, 4668291000, 125662750464, 3706032771336, 118759029538368, 4109063510399088, 152696171895135744, 6065376023980289424, 256455323932682550528, 11499944141042532006432, 545124523779848580648960

Offset

1,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Formula

E.g.f.: A(x) = Series_Reversion( exp(x) - exp(x^2) ).

a(n) = -sum(k=1..n-1, (k^n/n!+(k^(n/2)*(-1)^k*((-1)^n+1))/(2*(n/2)!)+sum(j=1..k-1, (-1)^(k-j)*binomial(k,j)*sum(m=1..n, (j^(2*m-n)*(k-j)^(n-m)*binomial(m,n-m))/m!)))*a(k)), n>1, a(1)=1. - Vladimir Kruchinin, Jun 25 2011

a(n) ~ n^(n-1) * sqrt((2*s-1)/(2-2*s+4*s^2)) / (exp(1+s)*(1-1/(2*s)))^n, where s = 0.6310764773894916166238... is the root of the equation exp(s) = 2*s*exp(s^2). - Vaclav Kotesovec, Jan 06 2014

Mathematica

Rest[CoefficientList[InverseSeries[Series[E^x-E^(x^2), {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 06 2014 *)

Prog

(PARI) {a(n)=if(n<1, 0, n!*polcoeff(serreverse(exp(x+x*O(x^n))-exp(x^2+x*O(x^n))), n))}
(Maxima)
  a(n):=if n=1 then 1 else -sum((k^n/n!+(k^(n/2)*(-1)^k*((-1)^n+1))/(2*(n/2)!)+sum((-1)^(k-j)*binomial(k, j)*sum((j^(2*m-n)*(k-j)^(n-m)*binomial(m, n-m))/m!, m, 1, n), j, 1, k-1))*a(k), k, 1, n-1); /* Vladimir Kruchinin, Jun 25 2011 */

Crossrefs

Sequence in context: A378378 A355408 A367384 * A206988 A217360 A371669

Adjacent sequences: A138011 A138012 A138013 * A138015 A138016 A138017

Keyword

nonn

Author

Paul D. Hanna, Feb 27 2008

Extensions

a(19)-a(20) from Vincenzo Librandi, Feb 20 2018

Status

approved