login
G.f. satisfies: A(x) = Sum_{n>=0} x^n / Product_{k=1..n} (1 - k*x*A(x)).
4

%I #5 Apr 11 2019 03:57:57

%S 1,1,2,6,22,92,424,2112,11236,63360,376800,2355016,15430784,105797968,

%T 757866592,5664174736,44109816528,357447744576,3010091812000,

%U 26304829992224,238217024498432,2232483865359488,21621812897089536,216130222764401024,2226983944005048960

%N G.f. satisfies: A(x) = Sum_{n>=0} x^n / Product_{k=1..n} (1 - k*x*A(x)).

%C It appears that all terms a(n) for n>1 are even.

%H Vaclav Kotesovec, <a href="/A225294/b225294.txt">Table of n, a(n) for n = 0..154</a>

%F G.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling2(n,k)*A(x)^(n-k).

%e G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 92*x^5 + 424*x^6 + 2112*x^7 +...

%e where

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

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

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}

%o {a(n)=local(A=1+x);for(i=0,n,A=sum(m=0,n,x^m*sum(k=0,m,Stirling2(m,k)*(A+x*O(x^n))^(m-k))));polcoeff(A,n)}

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

%Y Cf. A224922, A225293, A000110, A019538.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 04 2013