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 A111536 #16 Jul 12 2018 00:35:49
%S A111536 1,1,1,4,2,1,22,8,3,1,148,44,14,4,1,1156,296,84,22,5,1,10192,2312,600,
%T A111536 148,32,6,1,99688,20384,4908,1156,242,44,7,1,1069168,199376,44952,
%U A111536 10192,2084,372,58,8,1,12468208,2138336,454344,99688,20012,3528,544,74,9,1
%N A111536 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+2 of T), or [T^p](m,0) = p*T(p+m,p+2) for all m>=1 and p>=-2.
%C A111536 Column 0 equals A111529 (related to log of factorial series).
%C A111536 Column 2 (A111538) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A111541.
%H A111536 Paul Barry, <a href="https://arxiv.org/abs/1804.06801">A note on number triangles that are almost their own production matrix</a>, arXiv:1804.06801 [math.CO], 2018.
%F A111536 T(n, k) = k*T(n, k+1) + Sum_{j=0..n-k-1} T(j+1, 1)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+2, 1) = 2*T(n+1, 0), T(n+3, 3) = T(n+1, 0), for n>=0.
%e A111536 SHIFT_LEFT(column 0 of T^-2) = -2*(column 0 of T);
%e A111536 SHIFT_LEFT(column 0 of T^-1) = -1*(column 1 of T);
%e A111536 SHIFT_LEFT(column 0 of log(T)) = column 2 of T;
%e A111536 SHIFT_LEFT(column 0 of T^1) = 1*(column 3 of T);
%e A111536 SHIFT_LEFT(column 0 of T^2) = 2*(column 4 of T);
%e A111536 where SHIFT_LEFT of column sequence shifts 1 place left.
%e A111536 Triangle T begins:
%e A111536 1;
%e A111536 1, 1;
%e A111536 4, 2, 1;
%e A111536 22, 8, 3, 1;
%e A111536 148, 44, 14, 4, 1;
%e A111536 1156, 296, 84, 22, 5, 1;
%e A111536 10192, 2312, 600, 148, 32, 6, 1;
%e A111536 99688, 20384, 4908, 1156, 242, 44, 7, 1;
%e A111536 1069168, 199376, 44952, 10192, 2084, 372, 58, 8, 1;
%e A111536 12468208, 2138336, 454344, 99688, 20012, 3528, 544, 74, 9, 1; ...
%e A111536 ...
%e A111536 After initial term, column 1 is twice column 0.
%e A111536 Matrix inverse T^-1 = A111540 starts:
%e A111536 1;
%e A111536 -1, 1;
%e A111536 -2, -2, 1;
%e A111536 -8, -2, -3, 1;
%e A111536 -44, -8, -2, -4, 1;
%e A111536 -296, -44, -8, -2, -5, 1;
%e A111536 -2312, -296, -44, -8, -2, -6, 1;
%e A111536 -20384, -2312, -296, -44, -8, -2, -7, 1;
%e A111536 -199376, -20384, -2312, -296, -44, -8, -2, -8, 1; ...
%e A111536 where columns are all equal after initial terms;
%e A111536 compare columns of T^-1 to column 1 of T.
%e A111536 Matrix logarithm log(T) = A111541 is:
%e A111536 0;
%e A111536 1, 0;
%e A111536 3, 2, 0;
%e A111536 14, 5, 3, 0;
%e A111536 84, 22, 8, 4, 0;
%e A111536 600, 128, 36, 12, 5, 0;
%e A111536 4908, 896, 212, 58, 17, 6, 0;
%e A111536 44952, 7220, 1496, 360, 90, 23, 7, 0;
%e A111536 454344, 65336, 12128, 2652, 602, 134, 30, 8, 0;
%e A111536 5016768, 653720, 110288, 22320, 4736, 974, 192, 38, 9, 0; ...
%e A111536 compare column 0 of log(T) to column 2 of T.
%t A111536 T[n_, k_] := T[n, k] = If[n<k || k<0, 0, If[n == k, 1, If[n == k+1, n, k * T[n, k+1] + Sum[T[j+1, 1]*T[n, j+k+1], {j, 0, n-k-1}]]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 24 2017, adapted from PARI *)
%o A111536 (PARI) T(n,k)=if(n<k || k<0,0,if(n==k,1,if(n==k+1,n, k*T(n,k+1)+sum(j=0,n-k-1,T(j+1,1)*T(n,j+k+1)))))
%o A111536 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%Y A111536 Cf. A111537 (column 1), A111538 (column 2), A111539 (row sums), A111540 (matrix inverse), A111541 (matrix log); related tables: A111528, A104980, A111544, A111553.
%K A111536 nonn,tabl
%O A111536 0,4
%A A111536 _Paul D. Hanna_, Aug 06 2005