A262717 a(n) = (n-1)*binomial(3*n-2,n)/(2*n-1)+(n+1)*binomial(3*n,n)/(2*n+1)-binomial(3*n-1,n).

1, 0, 1, 6, 35, 208, 1260, 7752, 48279, 303600, 1924065, 12271350, 78676884, 506662016, 3275052040, 21238169904, 138111313215, 900331830048, 5881813095795, 38499031112850, 252423322176795, 1657580519271600, 10899847657028400, 71764700685918240

Offset

0,4

Links

Table of n, a(n) for n=0..23.

R. J. Mathar, derivation of the D-finite recurrence

Formula

G.f. A(x) = 1+B'(x)*(1-x/B(x))^2, where B(x)/x is g.f. of A001764.

a(n) = Sum_{k=1..n}(k*binomial(n-1,k-2)*binomial(2*n,n-k))/n, n>0, a(0)=1.

D-finite with recurrence 2*n*(n-2)*(2*n+1)*(n+2)*a(n) -3*(n-1)*(n+3)*(3*n-4)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Mar 12 2017

Mathematica

Join[{0}, Table[(n - 1) Binomial[3 n - 2, n]/(2 n - 1) + (n + 1)Binomial[3 n, n]/(2 n + 1) - Binomial[3 n - 1, n], {n, 30}]] (* Vincenzo Librandi, Sep 28 2015 *)

Prog

(Maxima)
A(x):=x*(2/sqrt(3*x))*sin((1/3)*asin(sqrt(27*x/4)));
taylor(diff(A(x), x, 1)*(1-x/A(x))^2, x, 0, 20);
(Magma) [0] cat [(n-1)*Binomial(3*n-2, n)/(2*n-1)+(n+1)*Binomial(3*n, n)/(2*n+1)-Binomial(3*n-1, n): n in [1..30]]; // Vincenzo Librandi, Sep 28 2015
(PARI) a(n) = (n-1)*binomial(3*n-2, n)/(2*n-1)+(n+1)*binomial(3*n, n)/(2*n+1)-binomial(3*n-1, n);
vector(50, n, a(n-1)) \\ Altug Alkan, Sep 28 2015

Crossrefs

Cf. A001764.

Sequence in context: A180033 A354134 A260770 * A144638 A291246 A117671

Adjacent sequences: A262714 A262715 A262716 * A262718 A262719 A262720

Keyword

nonn,easy

Author

Vladimir Kruchinin, Sep 28 2015

Status

approved