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Triangle read by rows: S_B(n,k) = "Type B" Stirling numbers of the second kind.
4

%I #21 Jan 20 2024 08:59:59

%S 2,2,5,2,15,14,2,35,84,43,2,75,350,430,142,2,155,1260,2795,2130,499,2,

%T 315,4214,15050,19880,10479,1850,2,635,13524,73143,149100,132734,

%U 51800,7193,2,1275,42350,334110,987042,1320354,854700,258948,29186,2,2555,130620,1466515,6038550,11390673,10878000,5394750,1313370,123109

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

%H Michael De Vlieger, <a href="/A085483/b085483.txt">Table of n, a(n) for n = 1..11325</a> (rows n = 1..150, flattened)

%H Eli Bagno and David Garber, <a href="https://arxiv.org/abs/2401.08365">Combinatorics of q,r-analogues of Stirling numbers of type B</a>, arXiv:2401.08365 [math.CO], 2024. See page 4.

%H Takao Komatsu, Eli Bagno, and David Garber, <a href="https://arxiv.org/abs/2209.06674">A q,r-analogue of poly-Stirling numbers of second kind with combinatorial applications</a>, arXiv:2209.06674 [math.CO], 2022.

%F 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.

%e 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}.

%e Triangle begins:

%e 2;

%e 2, 5;

%e 2, 15, 14;

%e 2, 35, 84, 43;

%e 2, 75, 350, 430, 142;

%e 2, 155, 1260, 2795, 2130, 499;

%e ...

%t 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 *)

%Y Cf. A008277, A005425.

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

%K nonn,tabl

%O 1,1

%A _James East_, Aug 15 2003