A145453 Exponential transform of binomial(n,3) = A000292(n-2).

1, 0, 0, 1, 4, 10, 30, 175, 1176, 7084, 42120, 286605, 2270180, 19213766, 166326524, 1497096055, 14374680880, 147259920760, 1582837679056, 17659771122969, 204674606377140, 2473357218561250, 31148510170120420, 407154732691440811, 5504706823227724904

Offset: 0

Author

Alois P. Heinz, Oct 10 2008

Keywords

• nonn

Comments

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 3 balls are seen at the top.

a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 3 labels.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..530 (terms 0..200 from Alois P. Heinz)
• N. J. A. Sloane, Transforms
• Index entries for sequences related to rooted trees

Crossrefs

3rd column of A145460, A143398.

Cf. A292889, A292910.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *binomial(j,3) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);

Mathematica

Table[Sum[BellY[n, k, Binomial[Range[n], 3]], {k, 0, n}], {n, 0, 25}] (* Vladimir Reshetnikov, Nov 09 2016 *)

Formula

E.g.f.: exp(exp(x)*x^3/3!).