OFFSET
0,3
COMMENTS
Radius of convergence of g.f. A(x) is |x| < 0.339332122592...
More generally, given
A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-t*x)^k then
A(x) = (1-t*x) / sqrt( (1-(t+1)*x)^2*(1+x^2) + (2*t-3)*x^2 - 2*t*(t-1)*x^3 ).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..400
Colin Defant, James Propp, Quantifying Noninvertibility in Discrete Dynamical Systems, arXiv:2002.07144 [math.CO], 2020.
FORMULA
G.f.: (1-x) / sqrt(1 - 4*x + 4*x^2 - 4*x^3 + 4*x^4).
a(n) ~ (1-r) / (2 * sqrt(Pi*n) * sqrt(1 - 3*r + 2*r^2 - r^3) * r^n), where r = 0.33933212259239... is the root of the equation 1-4*r+4*r^2-4*r^3+4*r^4 = 0. - Vaclav Kotesovec, Feb 17 2014
a(n) = Sum_{m=0..n} Sum_{k=0,n-m} C(m,k)^2*C(n-m-1,n-m-k). - Vladimir Kruchinin, Jan 16 2018
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 42*x^5 + 114*x^6 + 314*x^7 + ...
where the g.f. equals the series:
A(x) = 1 +
x*(1 + x/(1-x)) +
x^2*(1 + 2^2*x/(1-x) + x^2/(1-x)^2) +
x^3*(1 + 3^2*x/(1-x) + 3^2*x^2/(1-x)^2 + x^3/(1-x)^3) +
x^4*(1 + 4^2*x/(1-x) + 6^2*x^2/(1-x)^2 + 4^2*x^3/(1-x)^3 + x^4/(1-x)^4) +
x^5*(1 + 5^2*x/(1-x) + 10^2*x^2/(1-x)^2 + 10^2*x^3/(1-x)^3 + 5^2*x^4/(1-x)^4 + x^5/(1-x)^5) + ...
MATHEMATICA
CoefficientList[Series[(1-x)/Sqrt[1 - 4*x + 4*x^2 - 4*x^3 + 4*x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 17 2014 *)
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k/(1-x +x*O(x^n))^k )), n)}
for(n=0, 40, print1(a(n), ", "))
(Maxima)
a(n):=sum(sum(binomial(m, k)^2*binomial(n-m-1, n-m-k), k, 0, n-m), m, 0, n);
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 10 2012
STATUS
approved