OFFSET
0,2
FORMULA
G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} ((1+x)^n - x*A(x))^n / 2^(n+1).
(2) A(x) = Sum_{n>=0} (1+x)^(n^2) / (2 + x*A(x))^(n+1).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 25*x^2 + 526*x^3 + 15554*x^4 + 590575*x^5 + 27338911*x^6 + 1492079652*x^7 + 93776123520*x^8 + 6669479907548*x^9 + ...
such that
A(x) = 1/2 + ((1+x) - x*A(x))/2^2 + ((1+x)^2 - x*A(x))^2/2^3 + ((1+x)^3 - x*A(x))^3/2^4 + ((1+x)^4 - x*A(x))^4/2^5 + ((1+x)^5 - x*A(x))^5/2^6 + ...
also,
A(x) = 1/(2 + x*A(x)) + (1+x)/(2 + x*A(x))^2 + (1+x)^4/(2 + x*A(x))^3 + (1+x)^9/(2 + x*A(x))^4 + (1+x)^16/(2 + x*A(x))^5 + (1+x)^25/(2 + x*A(x))^6 + ...
MATHEMATICA
m = 20; A[_] = 0;
Do[A[x_] = Sum[((1+x)^n - x A[x])^n/2^(n+1), {n, 0, k^2}] + O[x]^k, {k, 1, m}];
CoefficientList[A[x], x] // Round (* Jean-François Alcover, Oct 01 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 31 2019
STATUS
approved