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

1, 6, 11, 30, 52, 126, 219, 510, 896, 2046, 3632, 8190, 14666, 32766, 59099, 131070, 237832, 524286, 956196, 2097150, 3841586, 8388606, 15425136, 33554430, 61908562, 134217726, 248377154, 536870910, 996183062, 2147483646, 3994427099, 8589934590, 16013066072

Offset: 2

Author

Alois P. Heinz, Apr 28 2017

Keywords

• nonn

Comments

a(n) is odd if and only if n = 2^k with k>0.

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..1000
• Wikipedia, Partition of a set

Crossrefs

Column k=2 of A285824.

Cf. A285853.

Programs

Maple

a:= n-> 2*add(binomial(n, k), k=1..n/2)-
        `if`(n::even, 3/2*binomial(n, n/2), 0):
seq(a(n), n=2..40);
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [0, 1, 6, 11][n],
      (9*(n-1)*(n-4)*a(n-1)+2*(3*n^2-16*n+6)*a(n-2)
      -36*(n-2)*(n-4)*a(n-3)+8*(n-3)*(3*n-10)*a(n-4))
      /((3*n-13)*n))
    end:
seq(a(n), n=2..40);

Mathematica

a[n_] := 2*Sum[Binomial[n, k], {k, 1, n/2}] - If[EvenQ[n], 3/2*Binomial[n, n/2], 0];
Table[a[n], {n, 2, 40}] (* Jean-François Alcover, May 26 2018, from Maple *)

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