A317350 - OEIS

%I #11 Aug 10 2018 12:06:39

%S 1,1,2,12,200,4160,99862,2767792,87200166,3076185774,120118928740,

%T 5144915483804,239932734849080,12106729328331780,657428964058944716,

%U 38239094075667233528,2372421500769940561658,156417910715313378830238,10923007991339600108590688,805475337677577620666606928,62550798567594006106067173708

%N G.f. satisfies: A(x) = Sum_{n>=0} ( (1+x)^n - A(x) )^n  / (2 - (1+x)^n*A(x))^(n+1).

%C G.f. A(x) = G(log(1+x)), where G(x) is the e.g.f. of A317355.

%H Paul D. Hanna, <a href="/A317350/b317350.txt">Table of n, a(n) for n = 0..200</a>

%F G.f. A(x) satisfies:

%F (1) A(x) = Sum_{n>=0} ( (1+x)^n - A(x) )^n  / (2 - (1+x)^n*A(x))^(n+1),

%F (2) A(x) = Sum_{n>=0} ( (1+x)^n + A(x) )^n  / (2 + (1+x)^n*A(x))^(n+1).

%F a(n) ~ c * d^n * n! / sqrt(n), where d = A317904 = 3.956184203026... and c = 0.14581304299... - _Vaclav Kotesovec_, Aug 07 2018

%e G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 200*x^4 + 4160*x^5 + 99862*x^6 + 2767792*x^7 + 87200166*x^8 + 3076185774*x^9 + 120118928740*x^10 + ...

%e such that A = A(x) satisfies

%e A(x) = 1/(2 - A)  +  ((1+x) - A)/(2 - (1+x)*A)^2  +  ((1+x)^2 - A)^2/(2 - (1+x)^2*A)^3  +  ((1+x)^3 - A)^3/(2 - (1+x)^3*A)^4  +  ((1+x)^4 - A)^4/(2 - (1+x)^4*A)^5  +  ((1+x)^5 - A)^5/(2 - (1+x)^5*A)^6 + ...

%e Also,

%e A(x) = 1/(2 + A)  +  ((1+x) + A)/(2 + (1+x)*A)^2  +  ((1+x)^2 + A)^2/(2 + (1+x)^2*A)^3  +  ((1+x)^3 + A)^3/(2 + (1+x)^3*A)^4  +  ((1+x)^4 + A)^4/(2 + (1+x)^4*A)^5  +  ((1+x)^5 + A)^5/(2 + (1+x)^5*A)^6 + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A = Vec( sum(m=0,#A, ( (1+x)^m - Ser(A) )^m  / (2 - (1+x)^m*Ser(A))^(m+1) ) ) ); A[n+1]}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A317351, A317355.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 02 2018