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%I #21 Jul 19 2016 11:12:08
%S 1,1,1,4,25,206,2060,23920,314065,4582300,73393490,1278859176,
%T 24073541260,486806278752,10525038917720,242318610557760,
%U 5919811842140945,152974724047702626,4169576527021400852
%N G.f. A(x) satisfies: Series_Reversion( x/A(x) ) / x = 2*A(x) - (1+x).
%C The g.f. for A120956 = Series_Reversion( x/A(x) ) / x = 2*A(x) - (1+x), where A(x) is the g.f. of this sequence.
%H Paul D. Hanna, <a href="/A120955/b120955.txt">Table of n, a(n) for n = 0..256</a>
%F G.f. A(x) satisfies:
%F (1) A( x*(2*A(x) - 1-x) ) = 2*A(x) - 1-x.
%F (2) A(x) = 2*A(x/A(x)) - 1 - x/A(x).
%F (3) A(x) = F(x/A(x)) and F(x) = A(x*F(x)) where F(x) = g.f. of A120956.
%F (4) A(x) = 1+x + Sum{n>=1} x^n * d^n/dx^n (A(x)-1)^(n+1) / (n+1)!. - _Paul D. Hanna_, Dec 27 2012
%F a(n) = A120956(n)/2 for n>=2.
%F a(n) = [x^n] A(x)^n / (2*n) for n>1; i.e., a(n) equals the coefficient of x^n in A(x)^n divided by 2*n.
%F a(n) ~ c * n^(n + 1/2 + log(2)) / (exp(n) * (log(2))^n), where c = 0.33794865962155... . - _Vaclav Kotesovec_, Aug 10 2014
%e A(x) = 1 + x + x^2 + 4*x^3 + 25*x^4 + 206*x^5 + 2060*x^6 +...
%e The g.f. of A120956 is:
%e series_reversion(x/A(x))/x = 1 + x + 2*x^2 + 8*x^3 + 50*x^4 + 412*x^5 +...
%e Compare terms to see that A120956(n) = 2*a(n) for n>=2.
%e The g.f. satisfies the series:
%e A(x) = 1+x + x*d/dx (A(x)-1)^2/2! + x^2*d^2/dx^2 (A(x)-1)^3/3! + x^3*d^3/dx^3 (A(x)-1)^4/4! + x^4*d^4/dx^4 (A(x)-1)^5/5! +...
%o (PARI) a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,t); A[ #A]=subst(Vec(serreverse(x/Ser(A)))[ #A],t,0));A[n+1]
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D
%o {a(n)=local(A=1+x*O(x^n)); for(i=1, n, A=1+x+sum(m=1, n, x^m*Dx(m, (A-1+x*O(x^n))^(m+1)/(m+1)!) )); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", ")) \\ _Paul D. Hanna_, Dec 27 2012
%Y Cf. A120956, A120970 (variant), A249933.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jul 19 2006
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