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A360861 a(n) = Sum_{k=0..n} binomial(n, ceiling(k/2)) * binomial(n, floor(k/2)).

%I A360861 #19 Dec 11 2023 05:43:12
%S A360861 1,2,7,22,81,281,1058,3830,14605,54127,208110,782761,3027038,11501478,
%T A360861 44668692,170974710,666220005,2564271875,10018268150,38728479647,
%U A360861 151631858378,588229029258,2307174835212,8975958379817,35258881445606,137501193282026,540821096592028
%N A360861 a(n) = Sum_{k=0..n} binomial(n, ceiling(k/2)) * binomial(n, floor(k/2)).
%H A360861 Paolo Xausa, <a href="/A360861/b360861.txt">Table of n, a(n) for n = 0..1000</a>
%F A360861 a(n) = (1/2)*(binomial(2*n+1,n)+binomial(n,floor(n/2))^2). - _Tani Akinari_, Jul 12 2023
%t A360861 A360861[n_]:=(Binomial[2n+1,n]+Binomial[n,Floor[n/2]]^2)/2;
%t A360861 Array[A360861,30,0] (* _Paolo Xausa_, Dec 11 2023 *)
%o A360861 (Python)
%o A360861 from math import comb
%o A360861 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 A360861 (Maxima) a(n):=(1/2)*(binomial(2*n+1,n)+(binomial(n,floor(n/2)))^2); /* _Tani Akinari_, Jul 12 2023 */
%Y A360861 Row sums of A360859.
%Y A360861 Cf. A001405, A018224.
%K A360861 nonn
%O A360861 0,2
%A A360861 _Peter Luschny_, Feb 28 2023