A272515 Number of set partitions of [n] into three blocks with distinct sizes.

60, 105, 448, 2016, 4980, 15675, 61644, 155155, 482573, 1733550, 4549808, 13890360, 48104628, 128949675, 392009140, 1322692581, 3607864403, 10929721440, 36245555284, 100109572875, 302709337515, 990788537700, 2763564406113, 8344789976616, 27039048750600

Offset: 6

Author

Alois P. Heinz, May 01 2016

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 6..1000

Crossrefs

Column k=3 of A131632.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
      or t*(2*i+1-t)/2n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
    end:
a:= n-> b(n$2, 3):
seq(a(n), n=6..40);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[t > i || t(t+1)/2 > n || t(2i+1-t)/2 < n, 0, If[n == 0, 1, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t - 1]* Binomial[n, i]]]];
a[n_] := b[n, n, 3];
a /@ Range[6, 40] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

Formula

a(n) = n! * [x^n*y^3] Product_{n>=1} (1+y*x^n/n!).

Conjecture: a(n) ~ 3^n / 6. - Vaclav Kotesovec, Dec 11 2020