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
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.
Programs
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Magma
[(n+3)*Catalan(n+1): n in [0..30]]; // Vincenzo Librandi, Sep 11 2016
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Maple
seq((n+3)*binomial(2*n+2, n+1)/(n+2), n=0..24); # Zerinvary Lajos, Dec 08 2008
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Mathematica
Table[(n + 3) (CatalanNumber[n + 1]), {n, 0, 30}] (* Vincenzo Librandi, Sep 11 2016 *)
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)