OFFSET
0,3
COMMENTS
Limit a(n)/a(n+1) = 0.1980622641951617475..., a root of 1 - 6*x + 5*x^2 - x^3 = 0.
FORMULA
G.f.: (1-x)^3 / sqrt((1 - 2*x + x^2 - x^3)*(1 - 6*x + 5*x^2 - x^3)).
G.f.: (1-x)^3 / sqrt(1 - 8*x + 18*x^2 - 18*x^3 + 13*x^4 - 6*x^5 + x^6).
EXAMPLE
G.f.: A(x) = 1 + x + 6*x^2 + 27*x^3 + 121*x^4 + 555*x^5 + 2586*x^6 +...
where
A(x) = 1 + x/(1-x)^4*(1+x) + x^2/(1-x)^8*(1+2^2*x+x^2) + x^3/(1-x)^12*(1+3^2*x+3^2*x^2+x^3) + x^4/(1-x)^16*(1+4^2*x+6^2*x^2+4^2*x^3+x^4) + x^5/(1-x)^20*(1+5^2*x+10^2*x^2+10^2*x^3+5^2*x^4+x^5) +...
PROG
(PARI) {a(n)=polcoeff( sum(m=0, n, x^m * sum(k=0, m, binomial(m, k)^2 * x^k) / (1-x +x*O(x^n))^(4*m)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=polcoeff( (1-x)^3 / sqrt(1 - 8*x + 18*x^2 - 18*x^3 + 13*x^4 - 6*x^5 + x^6 +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 12 2014
STATUS
approved