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A271930 G.f. A(x) satisfies: A(x) = A( x^2 + 6*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1. 4

%I #20 Apr 19 2016 17:59:45

%S 1,3,15,90,597,4221,31185,237897,1859568,14816637,119892942,982565883,

%T 8138777166,68028775587,573078135996,4860507197700,41470162208814,

%U 355695498901179,3065210379987489,26525947283576640,230425563258798840,2008561878414115803,17563090615911038115,154014411705019299450,1354142406561753259035,11934928413519024726252,105426063390991627937457,933206579920813459523994,8276480132736299734057275,73535083052134446419214960

%N G.f. A(x) satisfies: A(x) = A( x^2 + 6*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.

%C Compare the g.f. to the following related identities:

%C (1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2), where C(x) = x + C(x)^2 (A000108).

%C (2) F(x) = F( x^2 + 4*x*F(x)^2 )^(1/2), where F(x) = D(x)^2/x and D(x) = x + D(x)^3/x (A001764).

%H Paul D. Hanna, <a href="/A271930/b271930.txt">Table of n, a(n) for n = 1..300</a>

%F G.f. A(x) satisfies: A( x*G(x^2) - 3*x^2 ) = x, where G(x)^2 = G(x^2) + 6*x, and G(x) is the g.f. of A264412.

%F a(n) ~ c * d^n / n^(3/2), where d = 9.35010183959428615991060685319... and c = 0.0902227396498060205291555743... . - _Vaclav Kotesovec_, Apr 18 2016

%e G..f.: A(x) = x + 3*x^2 + 15*x^3 + 90*x^4 + 597*x^5 + 4221*x^6 + 31185*x^7 + 237897*x^8 + 1859568*x^9 + 14816637*x^10 + 119892942*x^11 + 982565883*x^12 +...

%e where A(x)^2 = A( x^2 + 6*x*A(x)^2 ).

%e RELATED SERIES.

%e A(x)^2 = x^2 + 6*x^3 + 39*x^4 + 270*x^5 + 1959*x^6 + 14724*x^7 + 113706*x^8 + 896994*x^9 + 7198257*x^10 + 58580766*x^11 + 482345937*x^12 + 4011023556*x^13 + 33637887441*x^14 +...

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

%e B(x) = x - 3*x^2 + 3*x^3 - 3*x^5 + 9*x^7 - 33*x^9 + 126*x^11 - 513*x^13 + 2214*x^15 - 9876*x^17 + 45045*x^19 - 209493*x^21 +...+ A264412(n)*x^(2*n+1) +...

%e such that B(x) = x*G(x^2) - 3*x^2 where G(x)^2 = G(x^2) + 6*x, and G(x) is the g.f. of A264412.

%o (PARI) {a(n) = my(A=x,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^2 + 6*X*A^2)^(1/2) ); polcoeff(A,n)}

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

%Y Cf. A264412, A271935, A271957, A271931, A271934.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Apr 16 2016

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