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

0,2

Links

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

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)).

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 *)

Prog

(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
(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
(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
(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

Crossrefs

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

Sequence in context: A358651 A128849 A294495 * A153741 A070207 A268559

Adjacent sequences: A188286 A188287 A188288 * A188290 A188291 A188292

Keyword

nonn

Author

Olivier Gérard, Aug 19 2012

Status

approved