A272516 Number of set partitions of [n] into four blocks with distinct sizes.

12600, 27720, 138600, 643500, 4408404, 12687675, 60780720, 238299880, 1295666424, 4208874756, 18840460800, 72351683460, 361100656224, 1228553894491, 5370616442928, 20605640103400, 97659853077800, 342942099783075, 1479570975628200, 5678915129142255

Offset: 10

Author

Alois P. Heinz, May 01 2016

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 10..1000

Crossrefs

Column k=4 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, 4):
seq(a(n), n=10..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, 4];
a /@ Range[10, 40] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

Formula

a(n) = n! * [x^n*y^4] Product_{n>=1} (1+y*x^n/n!).