A093126 - OEIS

%I #5 Aug 16 2016 21:17:44

%S 1,1,2,3,6,10,19,33,62,110,204,366,675,1219,2239,4059,7439,13518,

%T 24737,45018,82304,149924,273929,499290,911902,1662787,3036105,

%U 5537577,10109364,18441799,33663239,61416729,112099746,204536183,373305550,681166986,1243173492,2268490929,4140035734,7554756990,13787320832,25159612832,45915363672

%N G.f.: A(x) = x/(1 - x - G001190(x^2)), where G001190 is the g.f. of A001190, the Wedderburn-Etherington numbers (binary rooted trees).

%C Not the same as A003237.

%F G.f. satisfies the following identities:

%F (1) A(x^2) = A(x)^2 / (1 + 2*A(x) + 2*A(x)^2),

%F (2) A(-x) = -A(x) / (1 + 2*A(x)),

%F (3) A(x) + A(-x) = -2*A(x)*A(-x),

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

%e A(x) = x + x^2 + 2x^3 + 3x^4 + 6x^5 + 10x^6 + 19x^7 + 33x^8 + ... = x/(1-x -(x^2 + x^4 + x^6 + 2x^8 + 3x^10 + 11x^12 + 23x^14 + ...)).

%o (PARI) {a(n) = my(A=x,u,v); for(k=2,n,u=A+x*O(x^k); v=subst(u,x,x^2); A-=x^k*polcoeff(u^2 -v*(1+2*u+2*u^2),k+1)/2); polcoeff(A,n)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A001190, A003237.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Mar 23 2004

%E Changed offset to 1 and removed leading zero. - _Paul D. Hanna_, Aug 16 2016