A102366 - OEIS

%I #18 Oct 11 2022 18:49:44

%S 1,1,2,3,6,10,15,21,28,84,120,165,220,286,364,455,1820,2380,3060,3876,

%T 4845,5985,7315,8855,10626,53130,65780,80730,98280,118755,142506,

%U 169911,201376,237336,278256,324632,1947792,2324784,2760681,3262623,3838380,4496388

%N Number of subsets of {1,2,...,n} in which exactly half of the elements are less than or equal to sqrt(n).

%C Also number of subsets of [n] in which exactly half of the elements are squares: a(5) = 10: {}, {1,2}, {1,3}, {1,5}, {2,4}, {3,4}, {4,5}, {1,2,3,4}, {1,2,4,5}, {1,3,4,5}. - _Alois P. Heinz_, Oct 11 2022

%H Alois P. Heinz, <a href="/A102366/b102366.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = Sum_k C(floor(sqrt(n)),k)*C(n-floor(sqrt(n)),k) = A048093(n) + 1 = a(n-1) + A084919(n-1).

%F a(n) = binomial(n, floor(sqrt(n))). - _Paul D. Hanna_, Jun 25 2011

%e a(5) = 10 since the ten subsets of {1,2,3,4,5} are { }, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {1,2, 3,4}, {1,2, 3,5} and {1,2, 4,5}.

%o (PARI) {a(n)=if(n<0,0,binomial(n, sqrtint(n)))} /* _Paul D. Hanna_ */

%o (PARI) {a(n)=sum(k=0,sqrtint(n),binomial(sqrtint(n), k)*binomial(n-sqrtint(n),k))}

%Y Cf. A011782 for number of subsets with an even number of elements.

%Y Cf. A000290.

%K nonn

%O 0,3

%A _Henry Bottomley_, Feb 22 2005