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A370518 Triangle of numbers read by rows: T(n,k) = Sum_{i=0..n} binomial(n,i)*(n-i)!*Stirling1(i,k)*TC(n,i) where TC(n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+3,i+1)*(-1)^(i) for n >= 0, 0 <= k <= n.

%I A370518 #13 Feb 26 2024 10:07:23
%S A370518 1,-5,1,14,-9,1,-18,29,-12,1,0,-22,35,-14,1,0,-26,15,25,-15,1,0,-60,4,
%T A370518 75,-5,-15,1,0,-204,-56,259,70,-56,-14,1,0,-912,-484,1092,609,-168,
%U A370518 -126,-12,1,0,-5040,-3708,5480,4599,-231,-882,-210,-9,1,0,-33120,-30024,31820,36350,3675,-6027,-2370,-300,-5,1
%N A370518 Triangle of numbers read by rows: T(n,k) = Sum_{i=0..n} binomial(n,i)*(n-i)!*Stirling1(i,k)*TC(n,i) where TC(n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+3,i+1)*(-1)^(i) for n >= 0, 0 <= k <= n.
%C A370518 Generalized Stirling numbers of the first kind of the second order.
%H A370518 Igor Victorovich Statsenko, <a href="https://aeterna-ufa.ru/sbornik/IN-2024-02-2.pdf#page=15">On the ordinal numbers of triangles of generalized special numbers</a>, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.
%F A370518 T(n,k) = Sum_{i=0..n} binomial(n,i)*(n-i)!*Stirling1(i,k)*TC(m,n,i) where TC(m,n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+m+1,i+1)*(-1)^(i),m = 2 for n >= 0.
%e A370518 n\k     0     1     2     3     4     5     6
%e A370518 0:      1
%e A370518 1:     -5     1
%e A370518 2:     14    -9     1
%e A370518 3:    -18    29   -12     1
%e A370518 4:      0   -22    35   -14     1
%e A370518 5:      0   -26    15    25   -15     1
%e A370518 6:      0   -60     4    75    -5   -15     1
%p A370518 C:=(n,k)->n!/(k!*(n-k)!) : T0:=(m,n,k)->sum(C(n+1,n-k-p)*Stirling2(p+m+1,p+1)*((-1)^p), p=0..n-k) : T:=(m,n,k)->sum(C(n,r)*(n-r)!*Stirling1(r,k)*T0(m,n,r), r=0..n)  m:=2 : seq(seq T(m,n,k), k=0..n), n=0..10);
%Y A370518 For m=0 the formula gives the sequence A130534; for m=1 the formula gives the sequence A094645. In this case, we assume that A130534 consists of generalized Stirling numbers of the first kind of zero order, and A094645 consists of generalized Stirling numbers of the first kind of the first order.
%K A370518 tabl,sign
%O A370518 0,2
%A A370518 _Igor Victorovich Statsenko_, Feb 21 2024