A227726 a(n) = [x^n] (1 + x)/(1 - x)^(2*n+1).

1, 4, 20, 112, 660, 4004, 24752, 155040, 980628, 6249100, 40060020, 258048960, 1668903600, 10829900592, 70480305440, 459823234112, 3006465218196, 19694758782300, 129235131438140, 849311959095600, 5589126007740660, 36825913869817380, 242910890642347200

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

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

G.f.: (2*G(x) - 1) / (3 - 2*G(x)), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

2*n*(2*n - 1)*a(n) - 3*(3*n - 1)*(3*n - 2)*a(n-1)=0. - R. J. Mathar, Jul 28 2013

a(n) = Sum_{r = 0..n-1} C(n-1,r)*C(2*n,r) + Sum_{r = 0..n} C(n,r)*C(2*n,n + r) - J. M. Bergot, Mar 18 2014

From Peter Bala, Jul 12 2016: (Start)

For n >= 1, a(n) = 4*binomial(3*n - 1, n - 1) = 4*A025174(n).

a(n) = [x^n]( 1/C(-x)^4 )^n, where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. Cf. A224274. (End)

Example

G.f.: A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 660*x^4 + 4004*x^5 + ...
where a(n) equals the coefficient of x^n in (1+x)/(1-x)^(2*n+1)
and forms the main diagonal in the following table of coefficients:
(1+x)/(1-x)^1: [1, 2, 2, 2, 2, 2, 2, 2, 2, ...];
(1+x)/(1-x)^3: [1, 4, 9, 16, 25, 36, 49, 64, 81, ...];
(1+x)/(1-x)^5: [1, 6, 20, 50, 105, 196, 336, 540, ...];
(1+x)/(1-x)^7: [1, 8, 35, 112, 294, 672, 1386, 2640, ...];
(1+x)/(1-x)^9: [1, 10, 54, 210, 660, 1782, 4290, 9438, ...];
(1+x)/(1-x)^11:[1, 12, 77, 352, 1287, 4004, 11011, 27456, ...];
(1+x)/(1-x)^13:[1, 14, 104, 546, 2275, 8008, 24752, 68952, ...];
(1+x)/(1-x)^15:[1, 16, 135, 800, 3740, 14688, 50388, 155040, ...]; ...
Related series is G(x) = 1 + x*G(x)^3, which begins:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 +...+ A001764(n)*x^n +...

Mathematica

Join[{1}, Table[Binomial[3n, n]+Binomial[3n-1, n-1], {n, 30}]] (* Harvey P. Dale, Jan 15 2015 *)

Prog

(PARI) {a(n)=binomial(3*n, n)+binomial(3*n-1, n-1)}
(PARI) {a(n)=2*binomial(3*n-1, n) - 0^n}
(PARI) {a(n)=polcoeff((1+x)/(1-x+x*O(x^n))^(2*n+1), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A052227, A001764, A000108, A025174, A224274.

Sequence in context: A239643 A081335 A136783 * A080609 A003645 A081085

Adjacent sequences: A227723 A227724 A227725 * A227727 A227728 A227729

Keyword

nonn,easy

Author

Paul D. Hanna, Jul 22 2013

Status

approved