A360861 - OEIS

%I #19 Dec 11 2023 05:43:12

%S 1,2,7,22,81,281,1058,3830,14605,54127,208110,782761,3027038,11501478,

%T 44668692,170974710,666220005,2564271875,10018268150,38728479647,

%U 151631858378,588229029258,2307174835212,8975958379817,35258881445606,137501193282026,540821096592028

%N a(n) = Sum_{k=0..n} binomial(n, ceiling(k/2)) * binomial(n, floor(k/2)).

%H Paolo Xausa, <a href="/A360861/b360861.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = (1/2)*(binomial(2*n+1,n)+binomial(n,floor(n/2))^2). - _Tani Akinari_, Jul 12 2023

%t A360861[n_]:=(Binomial[2n+1,n]+Binomial[n,Floor[n/2]]^2)/2;

%t Array[A360861,30,0] (* _Paolo Xausa_, Dec 11 2023 *)

%o (Python)

%o from math import comb

%o def A360861(n): return sum(comb(n,m:=k>>1)**2*(n-m)//(m+1) for k in range(1,n+1,2)) + sum(comb(n,k>>1)**2 for k in range(0,n+1,2)) # _Chai Wah Wu_, Feb 28 2023

%o (Maxima) a(n):=(1/2)*(binomial(2*n+1,n)+(binomial(n,floor(n/2)))^2); /* _Tani Akinari_, Jul 12 2023 */

%Y Row sums of A360859.

%Y Cf. A001405, A018224.

%K nonn

%O 0,2

%A _Peter Luschny_, Feb 28 2023