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A341102 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.

%I A341102 #15 Feb 25 2021 16:18:53
%S A341102 1,5,4,23,35,10,119,243,135,20,719,1701,1323,385,35,5039,12941,12166,
%T A341102 5068,910,56,40319,109329,115099,59514,15498,1890,84,362879,1026065,
%U A341102 1163370,689575,226800,40446,3570,120,3628799,10627617,12725075,8263750,3170200,722568,93786,6270,165
%N A341102 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 A341102 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 A341102 T(n, k) = Sum_{j=0..k} (binomial(n+j-1, 2*k) - binomial(n+k-j, 2*k))*A340556(k, j).
%F A341102 E.g.f.: (1 - z)^(-x) - exp(x*(exp(z) - 1)) (unrestricted rows and n >= 0).
%e A341102 Triangle starts:
%e A341102 [ 3] [1]
%e A341102 [ 4] [5,      4]
%e A341102 [ 5] [23,     35,      10]
%e A341102 [ 6] [119,    243,     135,     20]
%e A341102 [ 7] [719,    1701,    1323,    385,    35]
%e A341102 [ 8] [5039,   12941,   12166,   5068,   910,    56]
%e A341102 [ 9] [40319,  109329,  115099,  59514,  15498,  1890,  84]
%e A341102 [10] [362879, 1026065, 1163370, 689575, 226800, 40446, 3570, 120]
%p A341102 # Giving full rows for n >= 0:
%p A341102 gf := (1 - z)^(-x) - exp(x*(exp(z) - 1));
%p A341102 ser := series(gf, z, 20): coeffz := n -> coeff(ser,z,n):
%p A341102 A341102row := n -> seq(n!*coeff(coeffz(n), x, k), k=0..n):
%p A341102 for n from 0 to 9 do A341102row(n) od;
%o A341102 (SageMath)
%o A341102 for n in (3..11):
%o A341102     print([stirling_number1(n, k) - stirling_number2(n, k) for k in (1..n-2)])
%o A341102 (PARI) T(n,k) = abs(stirling(n,k,1)) - stirling(n,k,2); \\ _Michel Marcus_, Feb 24 2021
%Y A341102 Cf. A132393, A048993, A048742, A033312, A000292, A340556.
%K A341102 nonn,tabl
%O A341102 3,2
%A A341102 _Peter Luschny_, Feb 24 2021