A162167 - OEIS

%I #2 Mar 30 2012 18:37:17

%S 1,1,5,61,945,18401,448033,13162689,452456833,17814732769,

%T 790829204481,39087256226129,2129136397634689,126740704552712433,

%U 8186173905423123457,570241787130949813441,42616054790603864116737

%N E.g.f. satisfies: A(x) = exp(x*exp(2*x*exp(3*x*A(x)))).

%C More generally, if A(x) = exp(p*x*exp(q*x*exp(r*x*A(x)))

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

%C a(n,m) = Sum_{k=0..n} Sum_{j=0..k} p^(n-k)*q^(k-j)*r^j*C(n,k)*C(k,j)*m*(j+m)^(n-k-1)*(n-k)^(k-j)*(k-j)^j.

%C ...

%C In general, if A(x) = F(x*G(x*H(x*A(x))) with F(0)=G(0)=H(0)=1,

%C where A(x)^m = Sum_{n>=0} a(n,m)*x^n, then

%C a(n,m) = Sum_{k=0..n} Sum_{j=0..k} {[x^(n-k)] F(x)^(j+m)*m/(j+m)} * {[x^(k-j)] G(x)^(n-k)} * {[x^j] H(x)^(k-j)}.

%C ...

%F a(n) = Sum_{k=0..n} Sum_{j=0..k} 2^(k-j)*3^j*C(n,k)*C(k,j)*(j+1)^(n-k-1)*(n-k)^(k-j)*(k-j)^j.

%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 61*x^3/3! + 945*x^4/4! + 18401*x^5/5! +...

%o (PARI) {a(n,m=1,p=1,q=2,r=3)=n!*sum(k=0,n,sum(j=0,k,p^(n-k)*q^(k-j)*r^j*m*(j+m)^(n-k-1)/(n-k)!*(n-k)^(k-j)/(k-j)!*(k-j)^j/j!))}

%Y Cf. A162161, A162162.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 27 2009

%E _Paul D. Hanna_, Jun 28 2009