A362925 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}.

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, 32, 1, 877, 877, 825, 706, 515, 275, 64, 1, 4140, 4140, 3937, 3479, 2744, 1785, 793, 128, 1, 21147, 21147, 20270, 18313, 15177, 11002, 6347, 2315, 256, 1, 115975, 115975, 111835, 102678, 88033, 68303, 45368, 23073, 6817, 512, 1

Offset: 0

Author

N. J. A. Sloane, Aug 10 2023, based on an email from Don Knuth

Keywords

• nonn
• tabl

Comments

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

Examples

			Triangle begins:
       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,    32,     1;
     877,    877,    825,    706,   515,   275,    64,     1;
    4140,   4140,   3937,   3479,  2744,  1785,   793,   128,    1;
   21147,  21147,  20270,  18313, 15177, 11002,  6347,  2315,  256,   1;
  115975, 115975, 111835, 102678, 88033, 68303, 45368, 23073, 6817, 512, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Row sums are A000110(n+1).

Columns k=0+1,2-5 give A000110, A078468(n-2) (for n>=2), A383052(n-3) (for n>=3), A383053(n-4) (for n>=4), A383054(n-5) (for n>=5).

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

T(2n,n) gives A367820.

Cf. A040027, A113547, A362924.

Programs

Maple

T:= (n, k)-> add(Stirling2(n-k, j)*(j+1)^k, j=0..n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 01 2023

Mathematica

A362925[n_, m_]:=Sum[StirlingS2[n-m,k](k+1)^m,{k,0,n-m}];
Table[A362925[n,m],{n,0,15},{m,0,n}] (* Paolo Xausa, Dec 04 2023 *)

Formula

Sum_{k=0..n} (k+1) * T(n,k) = A040027(n+1). - Alois P. Heinz, Dec 02 2023