A322737 - OEIS

%I #6 Aug 11 2021 17:13:13

%S 1,1,3,17,243,5041,122793,3433557,108824679,3857180303,151189425233,

%T 6495604450659,303671019221745,15353507145898735,835092643075565163,

%U 48637547540923032151,3020890094905581400107,199356631125403317760803,13932407051414083995444277,1028080194901048673942405547,79883891921410823861579965753

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

%H Paul D. Hanna, <a href="/A322737/b322737.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/(1-x)^n - A(x) )^n  / (2 - A(x)/(1-x)^n)^(n+1),

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

%F a(n) ~ c * A317904^n * n^n / exp(n), where c = 0.47061136383707... - _Vaclav Kotesovec_, Aug 11 2021

%e G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 243*x^4 + 5041*x^5 + 122793*x^6 + 3433557*x^7 + 108824679*x^8 + 3857180303*x^9 + 151189425233*x^10 + ...

%e such that A = A(x) satisfies

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

%e Also,

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

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

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

%Y Cf. A317350.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 24 2019