A111924 Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,1) for n >= 1.

1, 1, 1, 1, 3, 0, 1, 6, 3, 0, 1, 10, 15, 0, 0, 1, 15, 45, 15, 0, 0, 1, 21, 105, 105, 0, 0, 0, 1, 28, 210, 420, 105, 0, 0, 0, 1, 36, 378, 1260, 945, 0, 0, 0, 0, 1, 45, 630, 3150, 4725, 945, 0, 0, 0, 0, 1, 55, 990, 6930, 17325, 10395, 0, 0, 0, 0, 0, 1, 66, 1485, 13860, 51975, 62370

Offset

1,5

Comments

T(n,k) = number of partitions of an n-set into k nonempty subsets, each of size at most 2.

References

J. Y. Choi and J. D. H. Smith, On the unimodality and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.

Links

Table of n, a(n) for n=1..72.

Formula

The Choi-Smith reference gives many further properties and formulas.

T(n, k) = T(n-1, k-1) + (n-1)*T(n-2, k-1).

Example

Triangle begins:
1
1 1
1 3 0
1 6 3 0
1 10 15 0 0
1 15 45 15 0 0
1 21 105 105 0 0 0
1 28 210 420 105 0 0 0
1 36 378 1260 945 0 0 0 0

Mathematica

T[n_, 0] = 0; T[1, 1] = 1; T[2, 1] = 1; T[n_, k_] := T[n - 1, k - 1] + (n - 1)T[n - 2, k - 1]; Table[T[n, k], {n, 12}, {k, n, 1, -1}] // Flatten (* Robert G. Wilson v *)

Crossrefs

A100861 is another version of this triangle. Row sums give A000085.

Sequence in context: A247255 A105147 A335262 * A212880 A211510 A243984

Adjacent sequences: A111921 A111922 A111923 * A111925 A111926 A111927

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Nov 25 2005

Extensions

More terms from Robert G. Wilson v, Dec 09 2005

Status

approved