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A262072 Number T(n,k) of partitions of an n-set with distinct block sizes and maximal block size equal to k; triangle T(n,k), n>=0, ceiling((sqrt(1+8*n)-1)/2)<=k<=n, read by rows.

%I A262072 #16 Feb 04 2017 10:35:53
%S A262072 1,1,1,3,1,4,1,10,5,1,60,15,6,1,140,21,7,1,280,224,28,8,1,1260,630,
%T A262072 336,36,9,1,12600,3780,1050,480,45,10,1,34650,7392,1650,660,55,11,1,
%U A262072 110880,74844,12672,2475,880,66,12,1,360360,276276,140712,20592,3575,1144,78,13,1
%N A262072 Number T(n,k) of partitions of an n-set with distinct block sizes and maximal block size equal to k; triangle T(n,k), n>=0, ceiling((sqrt(1+8*n)-1)/2)<=k<=n, read by rows.
%H A262072 Alois P. Heinz, <a href="/A262072/b262072.txt">Rows n = 0..200, flattened</a>
%e A262072 Triangle T(n,k) begins:
%e A262072 : 1;
%e A262072 :    1;
%e A262072 :       1;
%e A262072 :       3,  1;
%e A262072 :           4,     1;
%e A262072 :          10,     5,    1;
%e A262072 :          60,    15,    6,    1;
%e A262072 :                140,   21,    7,   1;
%e A262072 :                280,  224,   28,   8,  1;
%e A262072 :               1260,  630,  336,  36,  9,  1;
%e A262072 :              12600, 3780, 1050, 480, 45, 10, 1;
%p A262072 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,`if`(n=0, 1,
%p A262072        b(n, i-1) +`if`(i>n, 0, binomial(n, i)*b(n-i, i-1))))
%p A262072     end:
%p A262072 T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):
%p A262072 seq(seq(T(n,k), k=ceil((sqrt(1+8*n)-1)/2)..n), n=0..14);
%t A262072 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]]; Table[T[n, k], {n, 0, 14}, {k, Ceiling[(Sqrt[1+8*n]-1)/2], n}] // Flatten (* _Jean-François Alcover_, Feb 04 2017, translated from Maple *)
%Y A262072 Row sums give A007837.
%Y A262072 Column sums give A262073.
%Y A262072 Cf. A002024, A262071, A262078 (same read by columns).
%K A262072 nonn,tabf
%O A262072 0,4
%A A262072 _Alois P. Heinz_, Sep 10 2015