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Triangle read by rows: T(n,m), n >= 0, 0 <= m <= n, is number of partitions of the set {1,2,...,n} that have at most one block contained in {1,...,m}.
4

%I #30 Dec 06 2023 14:36:56

%S 1,1,1,2,2,1,5,5,4,1,15,15,13,8,1,52,52,47,35,16,1,203,203,188,153,97,

%T 32,1,877,877,825,706,515,275,64,1,4140,4140,3937,3479,2744,1785,793,

%U 128,1,21147,21147,20270,18313,15177,11002,6347,2315,256,1,115975,115975,111835,102678,88033,68303,45368,23073,6817,512,1

%N Triangle read by rows: T(n,m), n >= 0, 0 <= m <= n, is number of partitions of the set {1,2,...,n} that have at most one block contained in {1,...,m}.

%C A variant of A113547 and A362924. See those entries for further information.

%H Alois P. Heinz, <a href="/A362925/b362925.txt">Rows n = 0..140, flattened</a>

%F Sum_{k=0..n} (k+1) * T(n,k) = A040027(n+1). - _Alois P. Heinz_, Dec 02 2023

%e Triangle begins:

%e 1;

%e 1, 1;

%e 2, 2, 1;

%e 5, 5, 4, 1;

%e 15, 15, 13, 8, 1;

%e 52, 52, 47, 35, 16, 1;

%e 203, 203, 188, 153, 97, 32, 1;

%e 877, 877, 825, 706, 515, 275, 64, 1;

%e 4140, 4140, 3937, 3479, 2744, 1785, 793, 128, 1;

%e 21147, 21147, 20270, 18313, 15177, 11002, 6347, 2315, 256, 1;

%e 115975, 115975, 111835, 102678, 88033, 68303, 45368, 23073, 6817, 512, 1;

%e ...

%p T:= (n, k)-> add(Stirling2(n-k, j)*(j+1)^k, j=0..n-k):

%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Dec 01 2023

%t A362925[n_, m_]:=Sum[StirlingS2[n-m,k](k+1)^m,{k,0,n-m}];

%t Table[A362925[n,m],{n,0,15},{m,0,n}] (* _Paolo Xausa_, Dec 04 2023 *)

%Y Row sums are A000110(n+1).

%Y Column k=0 gives A000110.

%Y Column k=2 gives A078468(n-2) for n>=2.

%Y T(n+j,n) give (for j=0-2): A000012, A000079, A007689.

%Y T(2n,n) gives A367820.

%Y Cf. A040027, A113547, A362924.

%K nonn,tabl

%O 0,4

%A _N. J. A. Sloane_, Aug 10 2023, based on an email from _Don Knuth_.