login
Expansion of exp(x)/(1 - x - x^2/2!), where a(n) = 1 + n*a(n-1) + C(n,2)*a(n-2).
3

%I #10 May 24 2013 03:55:19

%S 1,2,6,25,137,936,7672,73361,801705,9856342,134640146,2023140417,

%T 33163934641,588936102860,11263023492372,230783643185881,

%U 5044101110058737,117136294344278346,2880200768035996990

%N Expansion of exp(x)/(1 - x - x^2/2!), where a(n) = 1 + n*a(n-1) + C(n,2)*a(n-2).

%H Vincenzo Librandi, <a href="/A128230/b128230.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) ~ n!*exp(sqrt(3)-1)*((1+sqrt(3))/2)^(n+1)/sqrt(3) . - _Vaclav Kotesovec_, Oct 20 2012

%e E.g.f.: exp(x)/(1 - x - x^2/2!) = 1 + 2*x + 6*x^2/2! + 25*x^3/3! + 137*x^4/4! + 936*x^5/5! + 7672*x^6/6! +... + a(n)*x^n/n! +...

%e where a(n) = 1 + n*a(n-1) + n*(n-1)*a(n-2)/2.

%t CoefficientList[Series[E^x/(1-x-x^2/2), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 20 2012 *)

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

%Y Cf. A087214, A000629, A128231, A128232.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 20 2007