A161630 - OEIS

%I #22 Nov 16 2017 02:49:42

%S 1,1,3,19,181,2321,37501,731935,16758393,440525377,13077834841,

%T 432796650551,15799794395749,630773263606513,27339525297079269,

%U 1278550150117141231,64171287394646697841,3440711053857464325377

%N E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)) ).

%H G. C. Greubel, <a href="/A161630/b161630.txt">Table of n, a(n) for n = 0..373</a>

%H Vaclav Kotesovec, <a href="http://oeis.org/A245265/a245265.pdf">Asymptotic of sequences A161630, A212722, A212917 and A245265</a>

%F a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(n-1,n-k).

%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then

%F a(n,m) = Sum_{k=0..n} n! * m*(n-k+m)^(k-1)/k! * C(n-1,n-k).

%F E.g.f. satisfies: A(x) = exp(x) * A(x)^(x*A(x)). - _Paul D. Hanna_, Aug 02 2013

%F a(n) ~ n^(n-1) * (1+2*c)^(n+1/2) / (sqrt(1+c) * 2^(2*n+2) * exp(n) * c^(2*n+3/2)), where c = LambertW(1/2) = 0.351733711249195826... (see A202356). - _Vaclav Kotesovec_, Jan 10 2014

%e E.g.f: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...

%e log(A(x))/x = 1 + x*A(x) + x^2*A(x)^2 + x^3*A(x)^3 + x^4*A(x)^4 +...

%t Table[Sum[n! * (n-k+1)^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jan 10 2014 *)

%o (PARI) {a(n,m=1)=if(n==0,1,sum(k=0,n,n!/k!*m*(n-k+m)^(k-1)*binomial(n-1,n-k)))}

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

%Y Cf. A161633 (e.g.f. = log(A(x))/x).

%Y Cf. A212722, A212917, A245265, A125500.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 17 2009