%I #46 Feb 16 2022 06:23:15
%S 1,1,1,4,0,1,11,3,0,1,41,10,0,0,1,162,30,10,0,0,1,715,126,35,0,0,0,1,
%T 3425,623,56,35,0,0,0,1,17722,2934,364,126,0,0,0,0,1,98253,15165,2220,
%U 210,126,0,0,0,0,1,580317,86900,10560,330,462,0,0,0,0,0,1
%N Triangular array read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the shortest block has length k (1 <= k <= n).
%C Row sums are Bell numbers A000110.
%C Column 1 is A000296 (shifted).
%C From _Peter Luschny_, Apr 05 2011: (Start)
%C Sum_{k>1} T(n,k) = A000296(n) count the set partitions with blocks of size > 1.
%C T(n,1) = A000296(n-1) count the set partitions with blocks of size = 1. Thus for the Bell numbers A000110(n) = Sum_{k>=1} T(n,k) = A000296(n-1) + A000296(n). (End)
%H Alois P. Heinz, <a href="/A178979/b178979.txt">Rows n = 1..141, flattened</a>
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/SetPartitions">Set partitions</a>
%F E.g.f. for column k: exp((exp(x) - Sum_{i=0..k-1} x^i/i!)) - exp((exp(x) - Sum_{i=0..k} x^i/i!)).
%F From _Ludovic Schwob_, Jan 15 2022: (Start)
%F T(2n,n) = A001700(n) = C(2n-1,n) for n>0.
%F T(2n-1,n-1) = A001700(n) = C(2n-1,n) for n>1. (End)
%e T(4,2) = card ({12|34, 13|24, 14|23}) = 3. - _Peter Luschny_, Apr 05 2011
%e Triangle begins:
%e 1;
%e 1, 1;
%e 4, 0, 1;
%e 11, 3, 0, 1;
%e 41, 10, 0, 0, 1;
%e 162, 30, 10, 0, 0, 1;
%e 715, 126, 35, 0, 0, 0, 1;
%e ...
%p g := k-> exp(x)*(1-(GAMMA(k,x)/GAMMA(k))); egf := k-> exp(g(k))-exp(g(k+1));
%p T := (n,k)-> n!*coeff(series(egf(k), x, n+1), x, n):
%p seq(seq(T(n, k), k=1..n), n=1..9); # _Peter Luschny_, Apr 05 2011
%p # second Maple program:
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
%p add(b(n-i*j, i+1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
%p end:
%p T:= (n, k)-> b(n, k) -b(n, k+1):
%p seq(seq(T(n, k), k=1..n), n=1..12); # _Alois P. Heinz_, Mar 25 2016
%t a[k_]:= Exp[x]-Sum[x^i/i!,{i,0,k}]; Transpose[Table[Range[20]! Rest[CoefficientList[Series[Exp[a[k-1]]-Exp[a[k]],{x,0,20}],x]],{k,1,9}]]//Grid
%Y Cf. A080510, A145877, A000110, A000296.
%K nonn,tabl
%O 1,4
%A _Geoffrey Critzer_, Jan 02 2011