A085483 Triangle read by rows: S_B(n,k) = "Type B" Stirling numbers of the second kind.

2, 2, 5, 2, 15, 14, 2, 35, 84, 43, 2, 75, 350, 430, 142, 2, 155, 1260, 2795, 2130, 499, 2, 315, 4214, 15050, 19880, 10479, 1850, 2, 635, 13524, 73143, 149100, 132734, 51800, 7193, 2, 1275, 42350, 334110, 987042, 1320354, 854700, 258948, 29186, 2, 2555, 130620, 1466515, 6038550, 11390673, 10878000, 5394750, 1313370, 123109

Offset

1,1

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)

Eli Bagno and David Garber, Combinatorics of q,r-analogues of Stirling numbers of type B, arXiv:2401.08365 [math.CO], 2024. See page 4.

Takao Komatsu, Eli Bagno, and David Garber, A q,r-analogue of poly-Stirling numbers of second kind with combinatorial applications, arXiv:2209.06674 [math.CO], 2022.

Formula

A partition of {-n, ..., -1, 1, ..., n} into nonempty subsets X_1, ..., X_r is called "symmetric" if for each i -X_i = X_j for some j. S_B(n, k) is the number of such symmetric partitions whose induced partition on {1, ..., n} involves k nonempty subsets. S_B(n, k) = S(n, k) * a(k), where S(n, k) is A008277 and a(k) is A005425.

Example

S_B(2,2)=5 because the relevant partitions of {-2,-1,1,2} are: {-2|-1|1|2}, {-1,1|-2|2}, {-1|1|-2,2}, {-1,1|-2,2}, {1,-2|-1,2}.
Triangle begins:
  2;
  2,   5;
  2,  15,   14;
  2,  35,   84,   43;
  2,  75,  350,  430,  142;
  2, 155, 1260, 2795, 2130, 499;
  ...

Mathematica

nn = 10; f[n_] := Sum[2^(n - 3 k) n!/((n - 2 k)! k!), {k, 0, n}]; Do[f[n], {n, 0, nn}]; Table[f[k]*StirlingS2[n, k], {n, nn}, {k, n}] (* Michael De Vlieger, Sep 21 2022, after Robert G. Wilson v at A005425 *)

Crossrefs

Cf. A008277, A005425.

S_B(n, 1) + ... + S_B(n, n) = A002872(n).

Sequence in context: A192233 A144943 A114976 * A271654 A271622 A324505

Adjacent sequences: A085480 A085481 A085482 * A085484 A085485 A085486

Keyword

nonn,tabl

Author

James East, Aug 15 2003

Status

approved