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 A102316 #11 Jul 19 2024 19:05:18
%S A102316 1,1,1,1,3,3,1,7,16,16,1,15,63,127,127,1,31,220,728,1363,1363,1,63,
%T A102316 723,3635,10450,18628,18628,1,127,2296,16836,69086,180854,311250,
%U A102316 311250,1,255,7143,74487,419917,1505041,3683791,6173791,6173791,1,511,21940
%N A102316 Triangle, read by rows, where T(n,k) = T(n,k-1) + (k+1)*T(n-1,k) for n>k>0, T(n,0)=1 and T(n,n) = T(n,n-1) for n>=0.
%C A102316 Main diagonal is A082161 (with offset). Row sums give A102317.
%C A102316 T(n,k) = number of column-marked subdiagonal paths of steps east (1,0) and north (0,1) from the origin to (n,k). Subdiagonal means that the path never rises above the diagonal line y=x and column-marked means that for 1 <= i <= n, one unit square directly below the i-th east step and above the line y=-1 is marked. - _David Callan_, Feb 04 2006
%F A102316 T(n, k) = Sum_{j=0..k} (j+1)*T(n-1, j) for n>k>0, T(n, 0)=1 for n>=0. T(n, n) = A082161(n) for n>0; A082161(n+1) = Sum_{k=0..n} (k+1)*T(n, k).
%e A102316 T(5,2) = 220 = 1*1 + 2*15 + 3*63 = 1*T(4,0) + 2*T(4,1) + 3*T(4,2).
%e A102316 T(5,2) = 220 = 31 + 3*63 = T(5,1) + (2+1)*T(4,2).
%e A102316 T(5,3) = 728 = 220 + 4*127 = T(5,2) + (3+1)*T(4,3).
%e A102316 Rows begin:
%e A102316 [1],
%e A102316 [1,1],
%e A102316 [1,3,3],
%e A102316 [1,7,16,16],
%e A102316 [1,15,63,127,127],
%e A102316 [1,31,220,728,1363,1363],
%e A102316 [1,63,723,3635,10450,18628,18628],
%e A102316 [1,127,2296,16836,69086,180854,311250,311250],
%e A102316 [1,255,7143,74487,419917,1505041,3683791,6173791,6173791],...
%o A102316 (PARI) T(n,k)=if(n<k||k<0,0,if(n==0||k==0,1,T(n,k-1)+(k+1)*T(n-1,k)))
%Y A102316 Cf. A102086, A082161, A102317.
%K A102316 nonn,tabl
%O A102316 0,5
%A A102316 _Paul D. Hanna_, Jan 04 2005