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 A127058 #16 Jan 26 2025 18:41:29
%S A127058 1,2,2,10,6,3,74,42,12,4,706,414,108,20,5,8162,5058,1332,220,30,6,
%T A127058 110410,72486,19908,3260,390,42,7,1708394,1182762,342252,57700,6750,
%U A127058 630,56,8,29752066,21573054,6583788,1159700,138150,12474,952,72,9,576037442
%N A127058 Triangle, read by rows, defined by: T(n,k) = Sum_{j=0..n-k-1} T(j+k,k)*T(n-j,k+1) for n > k >= 0, with T(n,n) = n+1.
%C A127058 Column 0 is A000698, the number of shellings of an n-cube, divided by 2^n n!.
%C A127058 Column 1 is A115974, the number of Feynman diagrams of the proper self-energy at perturbative order n.
%H A127058 G. C. Greubel, <a href="/A127058/b127058.txt">Rows n = 0..15 of triangle, flattened</a>
%e A127058 Other recurrences exist, as shown by:
%e A127058 column 0 = A000698: T(n,0) = (2n+1)!! - Sum_{k=1..n} (2k-1)!!*T(n-k,0);
%e A127058 column 1 = A115974: T(n,1) = T(n+1,0) - Sum_{k=0..n-1} T(k,1)*T(n-k,0).
%e A127058 Illustrate the recurrence:
%e A127058 T(n,k) = Sum_{j=0..n-k-1} T(j+k,k)*T(n-j,k+1) (n > k >= 0)
%e A127058 at column k=1:
%e A127058 T(2,1) = T(1,1)*T(2,2) = 2*3 = 6;
%e A127058 T(3,1) = T(1,1)*T(3,2) + T(2,1)*T(2,2) = 2*12 + 6*3 = 42;
%e A127058 T(4,1) = T(1,1)*T(4,2) + T(2,1)*T(3,2) + T(3,1)*T(2,2) = 2*108 + 6*12 + 42*3 = 414;
%e A127058 at column k=2:
%e A127058 T(3,2) = T(2,2)*T(3,3) = 3*4 = 12;
%e A127058 T(4,2) = T(2,2)*T(4,3) + T(3,2)*T(3,3) = 3*20 + 12*4 = 108;
%e A127058 T(5,2) = T(2,2)*T(5,3) + T(3,2)*T(4,3) + T(4,2)*T(3,3) = 3*220 + 12*20 + 108*4 = 1332.
%e A127058 Triangle begins:
%e A127058 1;
%e A127058 2, 2;
%e A127058 10, 6, 3;
%e A127058 74, 42, 12, 4;
%e A127058 706, 414, 108, 20, 5;
%e A127058 8162, 5058, 1332, 220, 30, 6;
%e A127058 110410, 72486, 19908, 3260, 390, 42, 7;
%e A127058 1708394, 1182762, 342252, 57700, 6750, 630, 56, 8;
%e A127058 29752066, 21573054, 6583788, 1159700, 138150, 12474, 952, 72, 9; ...
%t A127058 T[n_,k_]:= If[k==n, n+1, Sum[T[j+k,k]*T[n-j,k+1], {j,0,n-k-1}]];
%t A127058 Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 03 2019 *)
%o A127058 (PARI) {T(n,k)=if(n==k,n+1,sum(j=0,n-k-1,T(j+k,k)*T(n-j,k+1)))}
%o A127058 (Sage)
%o A127058 def T(n, k):
%o A127058 if (k==n): return n+1
%o A127058 else: return sum(T(j+k,k)*T(n-j,k+1) for j in (0..n-k-1))
%o A127058 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jun 03 2019
%Y A127058 Columns: A000698, A115974, A127059.
%Y A127058 Row sums: A127060.
%Y A127058 Cf. A001147 ((2n-1)!!).
%K A127058 nonn,tabl
%O A127058 0,2
%A A127058 _Paul D. Hanna_, Jan 04 2007