%I #17 May 24 2018 11:08:17

%S 1,1,4,75,14301,40870872,2163410250576,2525542278491543715,

%T 75742007488274337351844747,66712890687959224726994385259183993,

%U 1942822997098466460791474215498474580001684381,2080073366817374333366496031890682227244159986035768679984

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

%C a(n)^(1/n^2) / sqrt(n) tends to exp(1/4)/sqrt(2) = 0.907943... . - _Vaclav Kotesovec_, May 14 2016

%H Alois P. Heinz, <a href="/A262073/b262073.txt">Table of n, a(n) for n = 0..36</a>

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

%p b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,`if`(n=0, 1,

%p b(n, i-1) +`if`(i>n, 0, binomial(n, i)*b(n-i, i-1))))

%p end:

%p T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):

%p a:= n-> add(T(k, n), k=n..n*(n+1)/2):

%p seq(a(n), n=0..14);

%t 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 T[n_, k_] := b[n, k] - If[k == 0, 0, b[n, k - 1]];

%t a[n_] := Sum[T[k, n], { k, n, n*(n + 1)/2}];

%t Table[a[n], {n, 0, 14}] (* _Jean-François Alcover_, May 24 2018, translated from Maple *)

%Y Column sums of A262072 or A262078.

%Y Cf. A000217.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 10 2015