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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).
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%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