A188289 Binomial sum related to rooted trees.

0, 2, 3, 14, 45, 167, 609, 2270, 8517, 32207, 122463, 467843, 1794195, 6903353, 26635773, 103020254, 399300165, 1550554583, 6031074183, 23493410759, 91638191235, 357874310213, 1399137067683, 5475504511859, 21447950506395, 84083979575117

Offset: 0

Author

Olivier Gérard, Aug 19 2012

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000984, A014300, A026641, A178792, A176479, A072547.

Programs

GAP

List([0..30], n-> Binomial(2*n,n) -(-1)^n -Sum([0..n-1], k-> Binomial(2*k,n-1))); # G. C. Greubel, Apr 29 2019

Magma

[n eq 0 select 0 else Binomial(2*n, n) -(-1)^n - (&+[Binomial(2*k, n-1): k in [0..n-1]]): n in [0..30]]; // G. C. Greubel, Apr 29 2019

Mathematica

Table[Binomial[2n,n]-(-1)^n-Sum[Binomial[2k,n-1],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Dec 10 2012 *)

PARI

{a(n) = binomial(2*n,n) -(-1)^n -sum(k=0,n-1, binomial(2*k,n-1))}; \\ G. C. Greubel, Apr 29 2019

Sage

[binomial(2*n,n) -(-1)^n -sum(binomial(2*k, n-1) for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, Apr 29 2019

Formula

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

a(n) = Sum_{k=1..n} binomial(n+k,k)*(Sum_{r=n-k..n} (-1)^r*binomial(n-k, r)).

a(n) = (-1)^n*2^(-(1+n))*(1 - 2^(1+n) + (-2)^n*binomial(2+2*n, 1+n) * hypergeometric2F1(1, 2+2*n; 2+n; -1)).