This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370516 #14 Feb 25 2024 15:37:25
%S A370516 1,-5,1,7,-4,1,-3,3,-3,1,0,0,0,-2,1,0,0,0,-2,-1,1,0,0,0,-2,-3,0,1,0,0,
%T A370516 0,-2,-5,-3,1,1,0,0,0,-2,-7,-8,-2,2,1,0,0,0,-2,-9,-15,-10,0,3,1,0,0,0,
%U A370516 -2,-11,-24,-25,-10,3,4
%N A370516 Triangle of numbers read by rows: T(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 A370516 Generalized binomial coefficients of the second order.
%H A370516 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 A370516 T(n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+m+1,i+1)*(-1)^i where m = 2 for n >= 0, 0 <= k <= n.
%e A370516 n\k 0 1 2 3 4 5 6
%e A370516 0: 1
%e A370516 1: -5 1
%e A370516 2: 7 -4 1
%e A370516 3: -3 3 -3 1
%e A370516 4: 0 0 0 -2 1
%e A370516 5: 0 0 0 -2 -1 1
%e A370516 6: 0 0 0 -2 -3 0 1
%p A370516 C:=(n,k)->n!/(k!*(n-k)!) : T:=(m,n,k)->sum(C(n+1,n-k-r)*Stirling2(r+m+1,r+1)*((-1)^r), r=0..n-k) : m:=2 : seq(seq T(m,n,k), k=0..n), n=0..10);
%Y A370516 For m=0 the formula gives the sequence A007318; for m=1 the formula gives the sequence A159854. In this case, we assume that A007318 consists of generalized binomial coefficients of order zero and A159854 consists of generalized binomial coefficients of order one.
%K A370516 tabl,sign
%O A370516 0,2
%A A370516 _Igor Victorovich Statsenko_, Feb 21 2024