A262768 - OEIS

%I #15 Sep 08 2022 08:46:14

%S 0,0,4,20,85,344,1365,5368,21021,82160,320892,1253240,4896034,

%T 19137104,74847175,292929840,1147223325,4496006880,17631691440,

%U 69189377400,271676550390,1067383059600,4195964793930,16503454480656,64943823784050,255687666536224

%N a(n) = binomial(2*n+2,n)-2*binomial(2*n,n)+binomial(2*n-2,n).

%F G.f.: (-2*x^3-3*x^2-2*x+1)/(2*x^2*sqrt(1-4*x))-1/(2*x^2)+1/2.

%F G.f. satisfies B'(x)*(1-x/B(x))^2, where B(x)+1 is g.f. of A000108.

%F a(n) = Sum_{k=0..n}(binomial(k,n-k)*2^(2*k-n)*binomial(n-1,k-2)).

%F 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

%t Table[Binomial[2 n + 2, n] - 2 Binomial[2 n, n] + Binomial[2 n - 2, n], {n, 0, 40}] (* _Vincenzo Librandi_, Oct 01 2015 *)

%o (Maxima)

%o B(x):=(1-sqrt(1-4*x))/(2*x)-1;

%o taylor(diff(B(x),x)*(1-x/B(x))^2,x,0,10);

%o (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

%o (PARI) a(n) = binomial(2*n+2,n)-2*binomial(2*n,n)+binomial(2*n-2,n);

%o vector (100, n, a(n-1)) \\ _Altug Alkan_, Oct 01 2015

%Y Cf. A000108.

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Sep 30 2015