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

%I A085483 #21 Jan 20 2024 08:59:59
%S A085483 2,2,5,2,15,14,2,35,84,43,2,75,350,430,142,2,155,1260,2795,2130,499,2,
%T A085483 315,4214,15050,19880,10479,1850,2,635,13524,73143,149100,132734,
%U A085483 51800,7193,2,1275,42350,334110,987042,1320354,854700,258948,29186,2,2555,130620,1466515,6038550,11390673,10878000,5394750,1313370,123109
%N A085483 Triangle read by rows: S_B(n,k) = "Type B" Stirling numbers of the second kind.
%H A085483 Michael De Vlieger, <a href="/A085483/b085483.txt">Table of n, a(n) for n = 1..11325</a> (rows n = 1..150, flattened)
%H A085483 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 A085483 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 A085483 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 A085483 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 A085483 Triangle begins:
%e A085483   2;
%e A085483   2,   5;
%e A085483   2,  15,   14;
%e A085483   2,  35,   84,   43;
%e A085483   2,  75,  350,  430,  142;
%e A085483   2, 155, 1260, 2795, 2130, 499;
%e A085483   ...
%t A085483 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 A085483 Cf. A008277, A005425.
%Y A085483 S_B(n, 1) + ... + S_B(n, n) = A002872(n).
%K A085483 nonn,tabl
%O A085483 1,1
%A A085483 _James East_, Aug 15 2003