A285918 Number of ordered set partitions of [n] into three blocks such that equal-sized blocks are ordered with increasing least elements.

1, 18, 75, 420, 1218, 4242, 14563, 42930, 132528, 432960, 1250340, 3814629, 12073701, 35074482, 106044555, 331913202, 967193328, 2917846758, 9062084298, 26507831559, 79848170823, 246771097680, 723922691700, 2178960263415, 6709005218503, 19728686792637

Offset: 3

Author

Alois P. Heinz, Apr 28 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..1000
• Wikipedia, Partition of a set

Crossrefs

Column k=3 of A285824.

Cf. A285854.

Programs

Maple

b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(x^j*b(n-i*j, i-1, p+j)*combinat
      [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)), x, 4)
    end:
a:= n-> coeff(b(n$2, 0), x, 3):
seq(a(n), n=3..30);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[x^j*b[n - i*j, i-1, p+j]*multinomial[n, Join[{n - i*j}, Table[i, j] ] ]/j!^2, {j, 0, n/i}]], {x, 0, 4}];
a[n_] := Coefficient[b[n, n, 0], x, 3];
Table[a[n], {n, 3, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)