A038665 Convolution of A007054 (super ballot numbers) with A000984 (central binomial coefficients).

3, 8, 25, 84, 294, 1056, 3861, 14300, 53482, 201552, 764218, 2912168, 11143500, 42791040, 164812365, 636438060, 2463251010, 9552774000, 37112526990, 144410649240, 562724141460, 2195581527360, 8576490341250, 33537507830424

Offset

0,1

Links

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

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Formula

a(n) = (n+3)*C(n+1) with C(n) the Catalan numbers A000108.

G.f.: c(x)*(4 - c(x))/sqrt(1 - 4*x) with c(x) the g.f. for the Catalan numbers.

From Amiram Eldar, May 16 2022: (Start)

Sum_{n>=0} 1/a(n) = 41/6 - 64*Pi/(9*sqrt(3)) + 2*Pi^2/3.

Sum_{n>=0} (-1)^n/a(n) = 57/10 - 256*log(phi)/(5*sqrt(5)) + 24*log(phi)^2, where phi is the golden ratio (A001622). (End)

Maple

seq((n+3)*binomial(2*n+2, n+1)/(n+2), n=0..24); # Zerinvary Lajos, Dec 08 2008

Mathematica

Table[(n + 3) (CatalanNumber[n + 1]), {n, 0, 30}] (* Vincenzo Librandi, Sep 11 2016 *)

Prog

(Magma) [(n+3)*Catalan(n+1): n in [0..30]]; // Vincenzo Librandi, Sep 11 2016

Crossrefs

Cf. A000108, A000777, A000984, A001622, A007054.

Sequence in context: A123638 A207651 A361959 * A172382 A151426 A045900

Adjacent sequences: A038662 A038663 A038664 * A038666 A038667 A038668

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved