A317351 - OEIS

%I #12 Aug 10 2018 12:06:34

%S 1,2,6,16,154,4584,130464,3816304,123180090,4422532004,175136909492,

%T 7585703878304,356923128965592,18139717839708536,990827454743868120,

%U 57910782633622271952,3607453763547725076028,238660376246383050751764,16714929289459273370819900,1235688614706272361317140840,96170725583233854961162923028

%N G.f. satisfies: A(x) = Sum_{n>=0} ( (1+x)^(n+1) - 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 A317356.

%H Paul D. Hanna, <a href="/A317351/b317351.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+1) - A(x) )^n  / (2 - (1+x)^n*A(x))^(n+1),

%F (2) A(x) = Sum_{n>=0} ( (1+x)^(n+1) + 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 = 0.23137523927... - _Vaclav Kotesovec_, Aug 07 2018

%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 16*x^3 + 154*x^4 + 4584*x^5 + 130464*x^6 + 3816304*x^7 + 123180090*x^8 + 4422532004*x^9 + 175136909492*x^10 + ...

%e such that A = A(x) satisfies

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

%e Also,

%e A(x) = 1/(2 + A)  +  ((1+x)^2 + A)/(2 + (1+x)*A)^2  +  ((1+x)^3 + A)^2/(2 + (1+x)^2*A)^3  +  ((1+x)^4 + A)^3/(2 + (1+x)^3*A)^4  +  ((1+x)^5 + A)^4/(2 + (1+x)^4*A)^5  +  ((1+x)^6 + 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+1) - Ser(A) )^m  / (2 - (1+x)^m*Ser(A))^(m+1) ) ) ); A[n+1]}

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

%Y Cf. A317350, A317356.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 02 2018