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 A127452 #9 Jun 14 2017 00:23:30
%S A127452 1,1,1,2,2,1,1,6,6,4,4,2,1,1,24,24,18,18,12,8,8,4,2,1,1,120,120,96,96,
%T A127452 72,54,54,36,24,16,16,8,4,2,1,1,720,720,600,600,480,384,384,288,216,
%U A127452 162,162,108,72,48,32,32,16,8,4,2,1,1
%N A127452 Triangle, read by rows of n*(n+1)/2 + 1 terms, generated by the recurrence: start with a single '1' in row 0; row n+1 is generated from row n by first inserting zeros at positions {(m+1)*(m+2)/2 - 1, m>=0} in row n and then taking the partial sums in reverse order.
%C A127452 The first column equals the factorials. Triangle A127420 is generated by a similar recurrence.
%H A127452 Paul D. Hanna, <a href="/A127452/b127452.txt">Rows n = 0..30, flattened.</a>
%F A127452 Sum_{k=0..n*(n+1)/2} k*T(n,k) = A018927(n+1) = Sum_{k=0..n} k*k!*{(k+1)^(n-k+1)-k^(n-k+1)}.
%F A127452 T(n,k) = (n-t)! * (n-t)^(k - t*(t+1)/2) * (n-t+1)^(t-k + t*(t+1)/2) where t=floor((sqrt(8*k+1)-1)/2). Also, Sum_{j=k*(k+1)/2..(k+1)*(k+2)/2-1} T(n,j) = A047969(n-k,k) = (n-k)!*((n-k+1)^(k+1)-(n-k)^(k+1)).
%e A127452 The triangle begins:
%e A127452 1;
%e A127452 1, 1;
%e A127452 2, 2, 1, 1;
%e A127452 6, 6, 4, 4, 2, 1, 1;
%e A127452 24, 24, 18, 18, 12, 8, 8, 4, 2, 1, 1;
%e A127452 120, 120, 96, 96, 72, 54, 54, 36, 24, 16, 16, 8, 4, 2, 1, 1;
%e A127452 720, 720, 600, 600, 480, 384, 384, 288, 216, 162, 162, 108, 72, 48, 32, 32, 16, 8, 4, 2, 1, 1;
%e A127452 ...
%e A127452 The recurrence is illustrated by the following examples.
%e A127452 Start with a single '1' in row 0.
%e A127452 To get row 1, insert 0 in row 0 at position 0,
%e A127452 and take partial sums in reverse order:
%e A127452 0,_1;
%e A127452 1,_1;
%e A127452 To get row 2, insert 0 in row 1 at positions [0,2],
%e A127452 and take partial sums in reverse order:
%e A127452 0,_1,_0,_1;
%e A127452 2,_2,_1,_1;
%e A127452 To get row 3, insert 0 in row 2 at positions [0,2,5],
%e A127452 and take partial sums in reverse order:
%e A127452 0,_2,_0,_2,_1,_0,_1;
%e A127452 6,_6,_4,_4,_2,_1,_1;
%e A127452 To get row 4, insert 0 in row 3 at positions [0,2,5,9],
%e A127452 and take partial sums in reverse order:
%e A127452 _0,__6,__0,__6,__4,_0,_4,_2,_1,_0,_1;
%e A127452 24,_24,_18,_18,_12,_8,_8,_4,_2,_1,_1;
%e A127452 etc.
%e A127452 Continuing in this way generates the factorials in the first column.
%o A127452 (PARI) T(n,k)=if(n<0 || k<0,0,if(n==0 && k==0,1, if(k==0, n!, if(issquare(8*k+1),T(n,k-1),T(n,k-1)-T(n-1,k-(sqrtint(8*k+1)+1)\2)))))
%o A127452 (PARI) T(n,k)=local(t=(sqrtint(8*k+1)-1)\2);(n-t)!*(n-t)^(k-t*(t+1)/2)*(n-t+1)^(t-k+t*(t+1)/2)
%Y A127452 Cf. A018927, A127420, A047969, A182961 (variant).
%K A127452 nonn,tabl
%O A127452 0,4
%A A127452 _Paul D. Hanna_, Jan 15 2007