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A285917 Number of ordered set partitions of [n] into two blocks such that equal-sized blocks are ordered with increasing least elements.

%I A285917 #18 May 26 2018 04:18:04
%S A285917 1,6,11,30,52,126,219,510,896,2046,3632,8190,14666,32766,59099,131070,
%T A285917 237832,524286,956196,2097150,3841586,8388606,15425136,33554430,
%U A285917 61908562,134217726,248377154,536870910,996183062,2147483646,3994427099,8589934590,16013066072
%N A285917 Number of ordered set partitions of [n] into two blocks such that equal-sized blocks are ordered with increasing least elements.
%C A285917 a(n) is odd if and only if n = 2^k with k>0.
%H A285917 Alois P. Heinz, <a href="/A285917/b285917.txt">Table of n, a(n) for n = 2..1000</a>
%H A285917 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A285917 a:= n-> 2*add(binomial(n, k), k=1..n/2)-
%p A285917         `if`(n::even, 3/2*binomial(n, n/2), 0):
%p A285917 seq(a(n), n=2..40);
%p A285917 # second Maple program:
%p A285917 a:= proc(n) option remember; `if`(n<5, [0, 1, 6, 11][n],
%p A285917       (9*(n-1)*(n-4)*a(n-1)+2*(3*n^2-16*n+6)*a(n-2)
%p A285917       -36*(n-2)*(n-4)*a(n-3)+8*(n-3)*(3*n-10)*a(n-4))
%p A285917       /((3*n-13)*n))
%p A285917     end:
%p A285917 seq(a(n), n=2..40);
%t A285917 a[n_] := 2*Sum[Binomial[n, k], {k, 1, n/2}] - If[EvenQ[n], 3/2*Binomial[n, n/2], 0];
%t A285917 Table[a[n], {n, 2, 40}] (* _Jean-François Alcover_, May 26 2018, from Maple *)
%o A285917 (PARI) a(n) = 2*sum(k=1, n\2, binomial(n, k)) - if (!(n%2), 3*binomial(n, n/2)/2); \\ _Michel Marcus_, May 26 2018
%Y A285917 Column k=2 of A285824.
%Y A285917 Cf. A285853.
%K A285917 nonn
%O A285917 2,2
%A A285917 _Alois P. Heinz_, Apr 28 2017