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A262078 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), k>=0, k<=n<=k*(k+1)/2, read by columns.

%I A262078 #17 May 09 2018 09:58:54
%S A262078 1,1,1,3,1,4,10,60,1,5,15,140,280,1260,12600,1,6,21,224,630,3780,
%T A262078 34650,110880,360360,2522520,37837800,1,7,28,336,1050,7392,74844,
%U A262078 276276,1513512,9459450,131171040,428828400,2058376320,9777287520,97772875200,2053230379200
%N A262078 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), k>=0, k<=n<=k*(k+1)/2, read by columns.
%H A262078 Alois P. Heinz, <a href="/A262078/b262078.txt">Columns k = 0..36, flattened</a>
%e A262078 Triangle T(n,k) begins:
%e A262078 : 1;
%e A262078 :    1;
%e A262078 :       1;
%e A262078 :       3,  1;
%e A262078 :           4,     1;
%e A262078 :          10,     5,    1;
%e A262078 :          60,    15,    6,    1;
%e A262078 :                140,   21,    7,   1;
%e A262078 :                280,  224,   28,   8,  1;
%e A262078 :               1260,  630,  336,  36,  9,  1;
%e A262078 :              12600, 3780, 1050, 480, 45, 10, 1;
%p A262078 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,`if`(n=0, 1,
%p A262078        b(n, i-1) +`if`(i>n, 0, binomial(n, i)*b(n-i, i-1))))
%p A262078     end:
%p A262078 T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):
%p A262078 seq(seq(T(n, k), n=k..k*(k+1)/2), k=0..7);
%t A262078 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], {k, 0, 7}, {n, k, k*(k+1)/2}] // Flatten (* _Jean-François Alcover_, Dec 18 2016, after _Alois P. Heinz_ *)
%Y A262078 Row sums give A007837.
%Y A262078 Column sums give A262073.
%Y A262078 Cf. A000217, A002024, A262071, A262072 (same read by rows).
%K A262078 nonn,tabf
%O A262078 0,4
%A A262078 _Alois P. Heinz_, Sep 10 2015