A158053 - OEIS

%I #2 Mar 30 2012 18:37:16

%S 1,1,3,9,37,183,1175,10405,132911,2533697,70988149,2886198771,

%T 168860266189,14046492509383,1668792185650203,280222608364043833,

%U 66930106539423614233,22572046654805538142763

%N G.f.: A(x) = exp( Sum_{n>=1} (1 + 2^n*x*A(x))^n * x^n/n ).

%C Compare to: M(x) = exp( Sum_{n>=1} (1 + x*M(x))^n * x^n/n ) where M(x) is the g.f. of the Motzkin numbers (A001006).

%e G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 37*x^4 + 183*x^5 + 1175*x^6 +...

%e log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 105*x^4/4 + 671*x^5/5 + 5525*x^6/6 +...

%e log(A(x)) = (1+2x*A(x))*x + (1+4x*A(x))^2*x^2/2 + (1+8x*A(x))^3*x^3/3 +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,(1+2^m*x*A)^m*x^m/m+x*O(x^n))));polcoeff(A,n)}

%Y Cf. A156214.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 11 2009