OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f.: Sum_{n>=0} (x + (1+x)^n)^n / (1+x + (1+x)^n)^(n+1).
G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (1+x - x*(1+x)^n)^(n+1).
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 33*x^3 + 362*x^4 + 5132*x^5 + 88997*x^6 + 1825144*x^7 + 43207267*x^8 + 1159596267*x^9 + 34790232210*x^10 + ...
such that
A(x) = 1/(1+x - x) + ((1+x) - 1)/(1+x - x*(1+x))^2 + ((1+x)^2 - 1)^2/(1+x - x*(1+x)^2)^3 + ((1+x)^3 - 1)^3/(1+x - x*(1+x)^3)^4 + ((1+x)^4 - 1)^4/(1+x - x*(1+x)^4)^5 + ((1+x)^5 - 1)^5/(1+x - x*(1+x)^5)^6 + ((1+x)^6 - 1)^6/(1+x - x*(1+x)^6)^7 + ...
Also,
A(x) = 1/(1+x + 1) + (x + (1+x))/(1+x + (1+x))^2 + (x + (1+x)^2)^2/(1+x + (1+x)^2)^3 + (x + (1+x)^3)^3/(1+x + (1+x)^3)^4 + (x + (1+x)^4)^4/(1+x + (1+x)^4)^5 + (x + (1+x)^5)^5/(1+x + (1+x)^5)^6 + (x + (1+x)^6)^6/(1+x + (1+x)^6)^7 + ...
PROG
(PARI) {a(n) = my(A = sum(m=0, n, ((1+x)^m - 1 +x*O(x^n))^m / (1+x - x*(1+x)^m +x*O(x^n))^(m+1)) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 03 2018
STATUS
approved