A104859 Partial sums of A001764.

1, 2, 5, 17, 72, 345, 1773, 9525, 52788, 299463, 1730178, 10144818, 60211926, 361042498, 2183809018, 13308564682, 81637319641, 503667864976, 3123298907641, 19456221197941, 121696331095636, 764008782313381

Offset: 0

Author

Emeric Deutsch, Apr 24 2005

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A001764.

Programs

Maple

a:=n->add(binomial(3*k,k)/(2*k+1),k=0..n): seq(a(n),n=0..26);

Mathematica

Table[Sum[Binomial[3k,k]/(2k+1),{k,0,n}],{n,0,20}] (* Emanuele Munarini, Apr 08 2011 *)

Maxima

makelist(sum(binomial(3*k,k)/(2*k+1),k,0,n),n,0,20); /* Emanuele Munarini, Apr 08 2011 */

Formula

a(n) = Sum_{k=0..n} binomial(3k, k)/(2k+1).

G.f.: T(z)/(1-z), where T = 1+z*T^3.

G.f.: 2*sin((1/3)*arcsin(sqrt(27*z/4)))/((1-z)*sqrt(3*z)).

Recurrence: 2*(2*n^2 + 9*n + 10)*a(n+2) - (31*n^2 + 99*n + 80)*a(1+n) + 3*(9*n^2 + 27*n + 20)*a(n) = 0. - Emanuele Munarini, Apr 08 2011

a(n) ~ 3^(3*n+7/2)/(23*sqrt(Pi)*2^(2*n+2)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^2 * A(x)^3. - Ilya Gutkovskiy, Jul 25 2021