OFFSET
2,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 2..201
FORMULA
G.f. A(x) = Sum_{n>=2} a(n)*x^n satisfies the following formulas.
(1) A(x - A(x)) = Sum_{n>=1} x^prime(n).
(2) A(x) = Sum_{n>=1} B(x)^prime(n) where B(x) = x + A(B(x)) and B(x - A(x)) = x.
EXAMPLE
G.f.: A(x) = x^2 + 3*x^3 + 14*x^4 + 97*x^5 + 801*x^6 + 7519*x^7 + 77362*x^8 + 854835*x^9 + 10008606*x^10 + 123019640*x^11 + ...
where
A(x - A(x)) = x^2 + x^3 + x^5 + x^7 + x^11 + x^13 + x^17 + ... + x^prime(n) + ...
also
A(x) = B(x)^2 + B(x)^3 + B(x)^5 + B(x)^7 + B(x)^11 + ... + B(x)^prime(n) + ...
where B(x) = x + A(B(x)) begins
B(x) = x + x^2 + 5*x^3 + 34*x^4 + 285*x^5 + 2712*x^6 + 28301*x^7 + 316987*x^8 + 3758998*x^9 + 46755855*x^10 + ...
Further,
B(B(x)) = x + 2*x^2 + 12*x^3 + 94*x^4 + 874*x^5 + 9061*x^6 + 101728*x^7 + 1214974*x^8 + 15260651*x^9 + 200003422*x^10 + ...
where
B(x) = x + B(B(x))^2 + B(B(x))^3 + B(B(x))^5 + B(B(x))^7 + ... + B(B(x))^prime(n) + ...
PROG
(PARI) \\ Generates N terms of this sequence \\
N = 30; A = x^2; B = x;
for(i=1, N, B = serreverse(x - A +x*O(x^N)); A = sum(n=1, N, B^prime(n)) ); Vec(A)
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 29 2023
STATUS
approved