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T(n, k) = [n, k] - {n, k}, where [n, k] are the (unsigned) Stirling cycle numbers and {n, k} the Stirling set numbers. Table T(n, k) read by rows, for n >= 3 and 1 <= k <= n-2.
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%I #15 Feb 25 2021 16:18:53

%S 1,5,4,23,35,10,119,243,135,20,719,1701,1323,385,35,5039,12941,12166,

%T 5068,910,56,40319,109329,115099,59514,15498,1890,84,362879,1026065,

%U 1163370,689575,226800,40446,3570,120,3628799,10627617,12725075,8263750,3170200,722568,93786,6270,165

%N T(n, k) = [n, k] - {n, k}, where [n, k] are the (unsigned) Stirling cycle numbers and {n, k} the Stirling set numbers. Table T(n, k) read by rows, for n >= 3 and 1 <= k <= n-2.

%H Peter Luschny, <a href="https://math.stackexchange.com/q/4037946">The difference of the Stirling cycle numbers and the Stirling set numbers</a>, Mathematics Stack Exchange, Feb. 2021.

%F T(n, k) = Sum_{j=0..k} (binomial(n+j-1, 2*k) - binomial(n+k-j, 2*k))*A340556(k, j).

%F E.g.f.: (1 - z)^(-x) - exp(x*(exp(z) - 1)) (unrestricted rows and n >= 0).

%e Triangle starts:

%e [ 3] [1]

%e [ 4] [5, 4]

%e [ 5] [23, 35, 10]

%e [ 6] [119, 243, 135, 20]

%e [ 7] [719, 1701, 1323, 385, 35]

%e [ 8] [5039, 12941, 12166, 5068, 910, 56]

%e [ 9] [40319, 109329, 115099, 59514, 15498, 1890, 84]

%e [10] [362879, 1026065, 1163370, 689575, 226800, 40446, 3570, 120]

%p # Giving full rows for n >= 0:

%p gf := (1 - z)^(-x) - exp(x*(exp(z) - 1));

%p ser := series(gf, z, 20): coeffz := n -> coeff(ser,z,n):

%p A341102row := n -> seq(n!*coeff(coeffz(n), x, k), k=0..n):

%p for n from 0 to 9 do A341102row(n) od;

%o (SageMath)

%o for n in (3..11):

%o print([stirling_number1(n, k) - stirling_number2(n, k) for k in (1..n-2)])

%o (PARI) T(n,k) = abs(stirling(n,k,1)) - stirling(n,k,2); \\ _Michel Marcus_, Feb 24 2021

%Y Cf. A132393, A048993, A048742, A033312, A000292, A340556.

%K nonn,tabl

%O 3,2

%A _Peter Luschny_, Feb 24 2021