A262073 Number of partitions of k-sets with distinct block sizes and maximal block size equal to n (n <= k <= n*(n+1)/2).

1, 1, 4, 75, 14301, 40870872, 2163410250576, 2525542278491543715, 75742007488274337351844747, 66712890687959224726994385259183993, 1942822997098466460791474215498474580001684381, 2080073366817374333366496031890682227244159986035768679984

Offset: 0

Author

Alois P. Heinz, Sep 10 2015

Keywords

• nonn

Comments

a(n)^(1/n^2) / sqrt(n) tends to exp(1/4)/sqrt(2) = 0.907943... . - Vaclav Kotesovec, May 14 2016

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..36

Crossrefs

Column sums of A262072 or A262078.

Cf. A000217.

Programs

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, binomial(n, i)*b(n-i, i-1))))
    end:
T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):
a:= n-> add(T(k, n), k=n..n*(n+1)/2):
seq(a(n), n=0..14);

Mathematica

b[n_, i_] := b[n, i] = If[i*(i + 1)/2 < n, 0, If[n == 0, 1, b[n, i - 1] + If[i > n, 0, Binomial[n, i]*b[n - i, i - 1]]]];
T[n_, k_] := b[n, k] - If[k == 0, 0, b[n, k - 1]];
a[n_] := Sum[T[k, n], { k, n, n*(n + 1)/2}];
Table[a[n], {n, 0, 14}] (* Jean-François Alcover, May 24 2018, translated from Maple *)

Formula

a(n) = Sum_{k=n..n*(n+1)/2} A262072(k,n).