A133611 -id:A133611 - cp's OEIS frontend

A211561 T(n,k) = number of nonnegative integer arrays of length n+k-1 with new values 0 upwards introduced in order, and containing the value k-1.

1, 1, 2, 1, 4, 5, 1, 7, 14, 15, 1, 11, 36, 51, 52, 1, 16, 81, 171, 202, 203, 1, 22, 162, 512, 813, 876, 877, 1, 29, 295, 1345, 3046, 4012, 4139, 4140, 1, 37, 499, 3145, 10096, 17866, 20891, 21146, 21147, 1, 46, 796, 6676, 29503, 72028, 106133, 115463, 115974, 115975

Offset: 1

R. H. Hardin, Apr 15 2012

Table starts
....1.....1......1......1.......1........1........1.........1..........1
....2.....4......7.....11......16.......22.......29........37.........46
....5....14.....36.....81.....162......295......499.......796.......1211
...15....51....171....512....1345.....3145.....6676.....13091......24047
...52...202....813...3046...10096....29503....77078....183074.....401337
..203...876...4012..17866...72028...256565...810470...2300949....5957407
..877..4139..20891.106133..503295..2134122..8016373..26869727...81381744
.4140.21146.115463.649045.3513522.17337685.76199007.298009584.1046405027
Reading along antidiagonals seems to create A137650. - R. J. Mathar, Nov 29 2015
See also A133611. - Alois P. Heinz, Aug 30 2019

			Some solutions for n=5, k=4:
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..1....1....1....0....1....1....1....1....0....1....1....1....1....1....1....0
..1....2....2....0....0....2....2....0....1....2....2....2....2....0....2....1
..2....0....2....0....2....0....3....2....2....2....3....3....2....2....0....2
..3....1....3....1....3....2....1....3....3....2....1....3....3....2....1....2
..4....0....3....0....3....3....4....1....3....3....0....2....4....3....2....2
..5....3....3....2....4....4....2....1....2....2....1....0....4....3....3....2
..2....0....1....3....5....4....4....4....4....2....0....4....3....1....2....3

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A000110.
Column 2 is A058692(n+1).
Column 3 is A058681(n+1).
Row 2 is A000124.
Cf. A133611, A137650.

Empirical: T(n,k) = Sum_{j=k..n+k-1} stirling2(n+k-1,j)

A137650 Triangle read by rows, A008277 * A000012.

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 15, 14, 7, 1, 52, 51, 36, 11, 1, 203, 202, 171, 81, 16, 1, 877, 876, 813, 512, 162, 22, 1, 4140, 4139, 4012, 3046, 1345, 295, 29, 1, 21147, 21146, 20891, 17866, 10096, 3145, 499, 37, 1, 115975, 115974, 115463

Offset: 1

Gary W. Adamson, Feb 01 2008

Left column = Bell numbers (A000110) starting (1, 2, 5, 15, 52, 203, ...). Row sums = A005493(n+1): (1, 3, 10, 37, 151, 674, ...).

Corresponding to the generalized Stirling number triangle of first kind A049444. - Peter Luschny, Sep 18 2011

			First few rows of the triangle are
    1;
    2,   1;
    5,   4,   1;
   15,  14,   7,   1;
   52,  51,  36,  11,   1;
  203, 202, 171,  81,  16,   1;
  877, 876, 813, 512, 162,  22,   1;
  ...

Cf. A000110, A008277, A005493, A049444.

A similar triangle is A133611.

A137650_row := proc(n) local k,i;
add(add(combinat[stirling2](n, n-i), i=0..k)*x^(n-k-1),k=0..n-1);
seq(coeff(%,x,k),k=0..n-1) end:
seq(print(A137650_row(n)),n=1..7); # Peter Luschny, Sep 18 2011

row[n_] := Table[StirlingS2[n, k], {k, 0, n}] // Reverse // Accumulate // Reverse // Rest;
Array[row, 10] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

A008277 * A000012 as infinite lower triangular matrices. Partial sums of A008277 rows starting from the right.

cp's OEIS Frontend

A211561 T(n,k) = number of nonnegative integer arrays of length n+k-1 with new values 0 upwards introduced in order, and containing the value k-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula