A322737 G.f. satisfies: A(x) = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n / (2 - A(x)/(1-x)^n)^(n+1).

1, 1, 3, 17, 243, 5041, 122793, 3433557, 108824679, 3857180303, 151189425233, 6495604450659, 303671019221745, 15353507145898735, 835092643075565163, 48637547540923032151, 3020890094905581400107, 199356631125403317760803, 13932407051414083995444277, 1028080194901048673942405547, 79883891921410823861579965753

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f. A(x) satisfies:

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

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

a(n) ~ c * A317904^n * n^n / exp(n), where c = 0.47061136383707... - Vaclav Kotesovec, Aug 11 2021

Example

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 + ...
such that A = A(x) satisfies
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 + ...
Also,
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 + ...

Prog

(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]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A317350.

Sequence in context: A072350 A181032 A331649 * A201107 A210902 A188803

Adjacent sequences: A322734 A322735 A322736 * A322738 A322739 A322740

Keyword

nonn

Author

Paul D. Hanna, Jan 24 2019

Status

approved