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

0,3

Links

Tilman Piesk, First 100 rows, flattened

R. Fray, A generating function associated with the generalized Stirling numbers, Fib. Quart. 5 (1967), 356-366.

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

Example

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

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 *)

Crossrefs

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

Row sums give A000258.

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

Sequence in context: A350510 A259332 A108767 * A193817 A227159 A294439

Adjacent sequences: A046814 A046815 A046816 * A046818 A046819 A046820

Keyword

tabl,nonn,easy,nice,changed

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson, Jan 13 2000

Status

approved