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A178979 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).

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