A140046 - OEIS

%I #2 Mar 30 2012 18:37:10

%S 1,1,3,10,41,186,922,4911,27830,166656,1049410,6922476,47698148,

%T 342483885,2557538781,19829608532,159393394129,1326509171669,

%U 11415703608635,101473987987073,930688926616454,8798656042121634

%N G.f. satisfies: A(x) = x/(1 - A(x+x^2)).

%e G.f.: A(x) = x + x^2 + 3*x^3 + 10*x^4 + 41*x^5 + 186*x^6 + 922*x^7 +...

%e A(x+x^2) = x + 2*x^2 + 5*x^3 + 20*x^4 + 90*x^5 + 454*x^6 + 2488*x^7 +...

%e Let B(x) = x + x^2; define B_{n+1}(x) = B( B_{n}(x) ) with B_0(x)=x;

%e then g.f. A(x) equals the continued fraction:

%e A(x) = x/(1 - B(x)/(1 - B_2(x)/(1 - B_3(x)/(1 - B_4(x)/(1 - ...)))))

%e where B_{n}(x) begin:

%e B_2(x) = x + 2*x^2 + 2*x^3 + x^4 ;

%e B_3(x) = x + 3*x^2 + 6*x^3 + 9*x^4 + 10*x^5 + 8*x^6 + 4*x^7 + x^8 ;

%e B_4(x) = x + 4*x^2 + 12*x^3 + 30*x^4 + 64*x^5 + 118*x^6 + 188*x^7 +...;

%e B_5(x) = x + 5*x^2 + 20*x^3 + 70*x^4 + 220*x^5 + 630*x^6 + 1656*x^7 +...

%o (PARI) {a(n)=local(A=x);if(n==0,A=x,for(i=1,n,A=x/(1-subst(A,x,x+x^2 +x*O(x^n))))); polcoeff(A,n)}

%Y Cf. A127782.

%K nonn

%O 1,3

%A _Paul D. Hanna_, May 09 2008