A262768 a(n) = binomial(2*n+2,n)-2*binomial(2*n,n)+binomial(2*n-2,n).

0, 0, 4, 20, 85, 344, 1365, 5368, 21021, 82160, 320892, 1253240, 4896034, 19137104, 74847175, 292929840, 1147223325, 4496006880, 17631691440, 69189377400, 271676550390, 1067383059600, 4195964793930, 16503454480656, 64943823784050, 255687666536224

Offset

0,3

Links

Table of n, a(n) for n=0..25.

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

Cf. A000108.

Sequence in context: A110154 A158608 A282084 * A250003 A343361 A320935

Adjacent sequences: A262765 A262766 A262767 * A262769 A262770 A262771

Keyword

nonn

Author

Vladimir Kruchinin, Sep 30 2015

Status

approved