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A259117 G.f. A(x) satisfies: A(x)^2 = A( x^2*(1+x)/(1-x) ), with A(0) = 0. 1

%I

%S 1,1,1,2,3,5,9,18,38,80,166,342,705,1463,3065,6492,13917,30217,66455,

%T 147920,332700,754544,1721596,3943344,9051155,20789895,47741587,

%U 109543782,251076670,574820868,1314663050,3004304786,6861785924,15668223170,35778071864,81723401730,186773111883,427182189689,977960176139

%N G.f. A(x) satisfies: A(x)^2 = A( x^2*(1+x)/(1-x) ), with A(0) = 0.

%C Radius of convergence is r = sqrt(2)-1, where r = r^2*(1+r)/(1-r), with A(r) = 1.

%C Compare to a g.f. M(x) of Motzkin numbers: M(x)^2 = M(x^2/(1-2*x)) where M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x).

%H Paul D. Hanna, <a href="/A259117/b259117.txt">Table of n, a(n) for n = 1..520</a>

%e G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 9*x^7 + 18*x^8 + 38*x^9 + 80*x^10 + 166*x^11 + 342*x^12 + 705*x^13 + 1463*x^14 + 3065*x^15 + 6492*x^16 +...

%e such that A(x^2*(1+x)/(1-x)) = A(x)^2, where

%e A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 11*x^6 + 20*x^7 + 38*x^8 + 76*x^9 + 159*x^10 + 338*x^11 + 719*x^12 + 1526*x^13 + 3235*x^14 + 6868*x^15 + 14638*x^16 +...

%e RELATED SERIES.

%e Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then

%e B(x) = x - x^2 + x^3 - 2*x^4 + 5*x^5 - 12*x^6 + 29*x^7 - 75*x^8 + 201*x^9 - 542*x^10 + 1480*x^11 - 4112*x^12 + 11562*x^13 - 32770*x^14 + 93593*x^15 +...

%e where

%e B(x^2) = B(x)^2*(1 + B(x))/(1 - B(x)).

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

%o for(n=1,50,print1(a(n),", "))

%K nonn

%O 1,4

%A _Paul D. Hanna_, Nov 08 2015

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