A046817 Triangle of generalized Stirling numbers of 2nd kind.

1, 1, 2, 1, 6, 5, 1, 12, 32, 15, 1, 20, 110, 175, 52, 1, 30, 280, 945, 1012, 203, 1, 42, 595, 3465, 8092, 6230, 877, 1, 56, 1120, 10010, 40992, 70756, 40819, 4140, 1, 72, 1932, 24570, 156072, 479976, 638423, 283944, 21147, 1, 90, 3120, 53550, 487704, 2350950, 5660615, 5971350

Offset: 0

Author

N. J. A. Sloane

Keywords

• tabl
• nonn
• easy
• nice

Examples

			Triangle begins:
      k = 0    1    2    3    4          sum
n
1         1                                1
2         1    2                           3
3         1    6    5                     12
4         1   12   32   15                60
5         1   20  110  175   52          358
		

Links

• Tilman Piesk, First 100 rows, flattened
• R. Fray, A generating function associated with the generalized Stirling numbers, Fib. Quart. 5 (1967), 356-366.

Crossrefs

Diagonals give A000558, A000559, A000110, A002378, etc.

Row sums give A000258.

Horizontal mirror triangle is A039810 (matrix square of Stirling2).

Programs

Mathematica

a[n_, k_] = Sum[StirlingS2[n, i]*StirlingS2[i, k], {i, k, n}]; Flatten[Table[a[n, k], {n, 1, 10}, {k, n, 1, -1}]][[1 ;; 53]]  (* Jean-François Alcover, Apr 26 2011 *)

Formula

a(n, k) = Sum_{i=k..n} S2(n, i)*S2(i, k).

E.g.f.: exp(exp(exp(x*y)-1)-1)^(1/y). - Vladeta Jovovic, Dec 14 2003

Extensions

More terms from David W. Wilson, Jan 13 2000