A221101 - OEIS

%I #6 Jan 16 2013 04:49:36

%S 1,1,2,12,144,3160,118380,7174188,692356896,104696597808,

%T 24680489921280,9010186432576560,5073501307520289600,

%U 4385657278007399474496,5797249519065509217375936,11674185903250032386477342880,35692663320428574506107140979200

%N E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(n*x))^n/n!.

%F  E.g.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(k*x)^n/n!.

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 144*x^4/4! + 3160*x^5/5! +...

%e where

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

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

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

%o (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}

%o {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*subst(A,x,k*x+x*O(x^n))^m)*x^m/m!)); n!*polcoeff(A, n)}

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

%Y Cf. A189981, A221100.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 01 2013