login
E.g.f. satisfies: A(x) = Sum{n>=0} x^n * A(n*x)/n!.
6

%I #18 Oct 16 2013 03:25:54

%S 1,1,3,16,149,2316,59047,2429554,159549945,16557985432,2693862309131,

%T 682199144788734,267277518618047797,161130714885281760100,

%U 148762112860064623199295,209444428223095096806228346,447998198975235291015396393713,1450973400598977755884988875863216

%N E.g.f. satisfies: A(x) = Sum{n>=0} x^n * A(n*x)/n!.

%F a(n) = Sum_{k=0..n-1} C(n,k)*(n-k)^k * a(k) for n>0 with a(0)=1.

%e A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 149*x^4/4! + 2316*x^5/5! +...

%e where

%e A(x) = 1 + x*A(x) + x^2*A(2*x)/2! + x^3*A(3*x)/3! + x^4*A(4*x)/4! + x^5*A(5*x)/5! +...

%e which leads to the recurrence illustrated by:

%e a(3) = 1*3^0*(1) + 3*2^1*(1) + 3*1^2*(3) = 16;

%e a(4) = 1*4^0*(1) + 4*3^1*(1) + 6*2^2*(3) + 4*1^3*(16) = 149;

%e a(5) = 1*5^0*(1) + 5*4^1*(1) + 10*3^2*(3) + 10*2^3*(16) + 5*1^4*(149) = 2316.

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

%o (PARI) {a(n)=local(A=1);for(i=1,n,A=sum(k=0,n,x^k/k!*subst(A,x,k*x)+x*O(x^n)));n!*polcoeff(A,n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A230323, A125282, A218683.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 29 2006, Sep 22 2007