OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n equals:
(1) Sum_{n>=0} (1+x)^n * ((1+2*x)^n + (1+x)^n)^n / (2*(1+x)^n + (1+2*x)^n)^(n+1).
(2) Sum_{n>=0} (1+x)^n * ((1+2*x)^n - (1+x)^n)^n / (2*(1+x)^n - (1+2*x)^n)^(n+1).
EXAMPLE
G.f.: A(x) = 1 + x + 5*x^2 + 47*x^3 + 641*x^4 + 11283*x^5 + 243755*x^6 + 6236425*x^7 + 184344339*x^8 + 6180934293*x^9 + ...
such that
A(x) = 1/3 + (1+x)*((1+x) + (1+2*x))/(2*(1+x) + (1+2*x))^2 + (1+x)^2*((1+x)^2 + (1+2*x)^2)^2/(2*(1+x)^2 + (1+2*x)^2)^3 + (1+x)^3*((1+x)^3 + (1+2*x)^3)^3/(2*(1+x)^3 + (1+2*x)^3)^4 + (1+x)^4*((1+x)^4 + (1+2*x)^4)^4/(2*(1+x)^4 + (1+2*x)^4)^5 + ...
Also,
A(x) = 1 + (1+x)*((1+2*x) - (1+x))/(2*(1+x) - (1+2*x))^2 + (1+x)^2*((1+2*x)^2 - (1+x)^2)^2/(2*(1+x)^2 - (1+2*x)^2)^3 + (1+x)^3*((1+2*x)^3 - (1+x)^3)^3/(2*(1+x)^3 - (1+2*x)^3)^4 + (1+x)^4*((1+2*x)^4 - (1+x)^4)^4/(2*(1+x)^4 - (1+2*x)^4)^5 + ...
PROG
(PARI) {a(n) = my(A=1, o=x*O(x^n)); A = sum(m=0, n, (1+x)^m*((1+2*x)^m - (1+x)^m +o)^m/(2*(1+x)^m - (1+2*x)^m +o)^(m+1)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 11 2018
STATUS
approved