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A188289 Binomial sum related to rooted trees. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A185895 A128849 A294495 * A153741 A070207 A268559

Adjacent sequences:  A188286 A188287 A188288 * A188290 A188291 A188292

KEYWORD

nonn

AUTHOR

Olivier Gérard, Aug 19 2012

STATUS

approved

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Last modified July 7 19:14 EDT 2020. Contains 335498 sequences. (Running on oeis4.)