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

%I A262073 #17 May 24 2018 11:08:17
%S A262073 1,1,4,75,14301,40870872,2163410250576,2525542278491543715,
%T A262073 75742007488274337351844747,66712890687959224726994385259183993,
%U A262073 1942822997098466460791474215498474580001684381,2080073366817374333366496031890682227244159986035768679984
%N A262073 Number of partitions of k-sets with distinct block sizes and maximal block size equal to n (n <= k <= n*(n+1)/2).
%C A262073 a(n)^(1/n^2) / sqrt(n) tends to exp(1/4)/sqrt(2) = 0.907943... . - _Vaclav Kotesovec_, May 14 2016
%H A262073 Alois P. Heinz, <a href="/A262073/b262073.txt">Table of n, a(n) for n = 0..36</a>
%F A262073 a(n) = Sum_{k=n..n*(n+1)/2} A262072(k,n).
%p A262073 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,`if`(n=0, 1,
%p A262073        b(n, i-1) +`if`(i>n, 0, binomial(n, i)*b(n-i, i-1))))
%p A262073     end:
%p A262073 T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):
%p A262073 a:= n-> add(T(k, n), k=n..n*(n+1)/2):
%p A262073 seq(a(n), n=0..14);
%t A262073 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 A262073 T[n_, k_] := b[n, k] - If[k == 0, 0, b[n, k - 1]];
%t A262073 a[n_] := Sum[T[k, n], { k, n, n*(n + 1)/2}];
%t A262073 Table[a[n], {n, 0, 14}] (* _Jean-François Alcover_, May 24 2018, translated from Maple *)
%Y A262073 Column sums of A262072 or A262078.
%Y A262073 Cf. A000217.
%K A262073 nonn
%O A262073 0,3
%A A262073 _Alois P. Heinz_, Sep 10 2015