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A358623 Regular triangle read by rows. T(n, k) = {{n, k}}, where {{n, k}} are the second order Stirling set numbers (or second order Stirling numbers). T(n, k) for 0 <= k <= n.

%I A358623 #9 Nov 26 2022 07:57:17
%S A358623 1,0,0,0,1,0,0,1,0,0,0,1,3,0,0,0,1,10,0,0,0,0,1,25,15,0,0,0,0,1,56,
%T A358623 105,0,0,0,0,0,1,119,490,105,0,0,0,0,0,1,246,1918,1260,0,0,0,0,0,0,1,
%U A358623 501,6825,9450,945,0,0,0,0,0,0,1,1012,22935,56980,17325,0,0,0,0,0,0
%N A358623 Regular triangle read by rows. T(n, k) = {{n, k}}, where {{n, k}} are the second order Stirling set numbers (or second order Stirling numbers). T(n, k) for 0 <= k <= n.
%C A358623 {{n, k}} are the number of k-quotient sets of an n-set having at least two elements in each equivalence class. This is the definition and notation (doubling the stacked delimiters of the Stirling set numbers) as given by Fekete (see link).
%C A358623 The formal definition expresses the second order Stirling set numbers as a binomial sum over second order Eulerian numbers (see the first formula below). The terminology 'associated Stirling numbers of second kind' used elsewhere should be dropped in favor of the more systematic one used here.
%C A358623 Also the Bell transform of sign(n) for n >= 0. For the definition of the Bell transform see A264428.
%D A358623 Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, Addison-Wesley, Reading, 2nd ed. 1994, thirty-fourth printing 2022.
%H A358623 Antal E. Fekete, <a href="http://www.jstor.org/stable/2974533">Apropos two notes on notation</a>, Amer. Math. Monthly, 101 (1994), 771-778.
%F A358623 T(n, k) = Sum_{j=0..k} (-1)^(k - j)*binomial(j, k - j)*<<n - k + j, j>>, where <<n, k>> denote the second order Eulerian numbers (extending Knuth's notation).
%F A358623 T(n, k) = n!*[z^k][t^n] exp(z*(exp(t) - t - 1)).
%F A358623 T(n, k) = Sum_{j=0..k} (-1)^(k - j)*binomial(n, k - j)*{n - k + j, j}, where {n, k} denotes the Stirling set numbers.
%F A358623 T(n, k) = (n - 1) * T(n-2, k-1) + k * T(n-1, k) with suitable boundary conditions.
%F A358623 T(n + k, k) = A269939(n, k), which might be called the Ward set numbers.
%e A358623 Triangle T(n, k) starts:
%e A358623 [0] 1;
%e A358623 [1] 0, 0;
%e A358623 [2] 0, 1,   0;
%e A358623 [3] 0, 1,   0,    0;
%e A358623 [4] 0, 1,   3,    0,    0;
%e A358623 [5] 0, 1,  10,    0,    0,  0;
%e A358623 [6] 0, 1,  25,   15,    0,  0,  0;
%e A358623 [7] 0, 1,  56,  105,    0,  0,  0,  0;
%e A358623 [8] 0, 1, 119,  490,  105,  0,  0,  0,  0;
%e A358623 [9] 0, 1, 246, 1918, 1260,  0,  0,  0,  0,  0;
%p A358623 T := (n, k) -> add(binomial(n, k - j)*Stirling2(n - k + j, j)*(-1)^(k - j),
%p A358623 j = 0..k): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
%p A358623 # Using the e.g.f.:
%p A358623 egf := exp(z*(exp(t) - t - 1)): ser := series(egf, t, 12):
%p A358623 seq(print(seq(n!*coeff(coeff(ser, t, n), z, k), k = 0..n)), n = 0..9);
%p A358623 # Using second order Eulerian numbers:
%p A358623 A358623 := proc(n, k) if n = 0 then return 1 fi;
%p A358623 add(binomial(j, n - 2*k)*combinat:-eulerian2(n - k, n - k - j - 1), j = 0..n-k-1)
%p A358623 end: seq(seq(A358623(n, k), k = 0..n), n = 0..11);
%o A358623 (Python) # recursion over rows
%o A358623 from functools import cache
%o A358623 @cache
%o A358623 def StirlingSetOrd2(n: int) -> list[int]:
%o A358623     if n == 0: return [1]
%o A358623     if n == 1: return [0, 0]
%o A358623     rov: list[int] = StirlingSetOrd2(n - 2)
%o A358623     row: list[int] = StirlingSetOrd2(n - 1) + [0]
%o A358623     for k in range(1, n // 2 + 1):
%o A358623         row[k] = (n - 1) * rov[k - 1] + k * row[k]
%o A358623     return row
%o A358623 for n in range(9): print(StirlingSetOrd2(n))
%o A358623 # Alternative, using function BellMatrix from A264428.
%o A358623 def f(k: int) -> int:
%o A358623     return 1 if k > 0 else 0
%o A358623 print(BellMatrix(f, 9))
%Y A358623 A008299 is an irregular subtriangle with more information.
%Y A358623 A358622 (second order Stirling cycle numbers).
%Y A358623 Cf. A000296 (row sums), alternating row sums (apart from sign): A000587, A293037, and A014182.
%Y A358623 Cf. A048993, A201637, A264428, A269939, A340264.
%K A358623 nonn,tabl
%O A358623 0,13
%A A358623 _Peter Luschny_, Nov 25 2022