OFFSET
0,3
FORMULA
G.f.: (-2*x^3-3*x^2-2*x+1)/(2*x^2*sqrt(1-4*x))-1/(2*x^2)+1/2.
G.f. satisfies B'(x)*(1-x/B(x))^2, where B(x)+1 is g.f. of A000108.
a(n) = Sum_{k=0..n}(binomial(k,n-k)*2^(2*k-n)*binomial(n-1,k-2)).
Conjecture: -2*(n+2)*(49*n-95)*a(n) +325*(n-1)*(n+1)*a(n-1) +2*(67*n-38) *(2*n-5)*a(n-2)=0. - R. J. Mathar, Jun 07 2016
MATHEMATICA
Table[Binomial[2 n + 2, n] - 2 Binomial[2 n, n] + Binomial[2 n - 2, n], {n, 0, 40}] (* Vincenzo Librandi, Oct 01 2015 *)
PROG
(Maxima)
B(x):=(1-sqrt(1-4*x))/(2*x)-1;
taylor(diff(B(x), x)*(1-x/B(x))^2, x, 0, 10);
(Magma) [Binomial(2*n+2, n)-2*Binomial(2*n, n)+Binomial(2*n-2, n): n in [0..30]]; // Vincenzo Librandi, Oct 01 2015
(PARI) a(n) = binomial(2*n+2, n)-2*binomial(2*n, n)+binomial(2*n-2, n);
vector (100, n, a(n-1)) \\ Altug Alkan, Oct 01 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Sep 30 2015
STATUS
approved