A298700 - OEIS

%I #20 Dec 29 2021 16:13:48

%S 1,6,25,120,581,2877,14421,72996,372229,1909336,9840909,50923041,

%T 264391973,1376654747,7185811685,37589283916,197005160825,

%U 1034244838815,5437798710585,28629290831670,150913830095445,796396974477495,4206974157985845,22243990866224505

%N a(n) = (n/2)*Sum_{k=1..n} C(n + k, n)*C(k, n - k)/k.

%H Harvey P. Dale, <a href="/A298700/b298700.txt">Table of n, a(n) for n = 1..1000</a>

%p a := n -> (n/2)*add(binomial(n + k, n)*binomial(k, n - k)/k, k=1..n):

%p seq(a(n), n=1..24);

%p # Alternatively:

%p a := n -> `if`(n mod 2=0, 1, n/2)*binomial(2*n - floor(n/2), ceil(n/2))*hypergeom(

%p [-floor(n/2), ceil(n/2), floor(3*(n+1)/2)], [n mod 2+1/2, ceil(n/2)+1], -1/4):

%p seq(simplify(a(n)), n=1..24);

%t Table[n/2 Sum[Binomial[n+k,n] Binomial[k,n-k]/k,{k,n}],{n,30}] (* _Harvey P. Dale_, Dec 29 2021 *)

%o (PARI) a(n) = (n/2)*sum(k=1, n, binomial(n+k, n)*binomial(k, n-k)/k); \\ _Michel Marcus_, Jan 27 2018

%K nonn

%O 1,2

%A _Peter Luschny_, Jan 26 2018