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 A116925 #23 Apr 01 2021 09:43:11
%S A116925 1,1,2,1,2,3,1,2,4,4,1,2,5,8,5,1,2,6,14,16,6,1,2,7,22,42,32,7,1,2,8,
%T A116925 32,92,132,64,8,1,2,9,44,177,422,429,128,9,1,2,10,58,310,1122,2074,
%U A116925 1430,256,10,1,2,11,74,506,2606,7898,10754,4862,512,11,1,2,12,92,782,5462,25202,60398,58202,16796,1024,12
%N A116925 Triangle read by rows: row n (n >= 0) consists of the elements g(i, n-i) (0 <= i <= n), where g(r,s) = 1 + Sum_{k=1..r} Product_{i=0..k-1} binomial(r+s-1, s+i) / binomial(r+s-1, i).
%C A116925 A generalized Catalan number triangle.
%C A116925 An alternative construction of this triangle. Begin with the Pascal triangle array, written as:
%C A116925 1 1 1 1 1 1 ...
%C A116925 1 2 3 4 5 6 ...
%C A116925 1 3 6 10 15 21 ...
%C A116925 1 4 10 20 35 56 ...
%C A116925 1 5 15 35 70 126 ...
%C A116925 ...
%C A116925 For each row r (r >= 0) in the above array, construct a triangle U(r) by applying the operation H defined below.
%C A116925 Then the r-th diagonal from the right in the new triangle is given by the row sums of U(r).
%C A116925 To define H, let us use row r=2, {1 3 6 10 15 ...}, as an illustration.
%C A116925 To get the 4th entry, take the first 4 terms of the row, reverse them and write them under the first 4 terms:
%C A116925 A: 1 3 6 10
%C A116925 B: 10 6 3 1
%C A116925 and form a new row C by beginning with 1 and iterating the map C' = C*B/A until we reach 1:
%C A116925 C: 1 10 20 10 1
%C A116925 E.g., 20 = (6 *10) / 3.
%C A116925 The sum of the terms {1 10 20 10 1} is 42, which is the 4th entry in the r=2 diagonal of the new triangle.
%C A116925 The full triangle U(2) begins
%C A116925 1
%C A116925 1 1
%C A116925 1 3 1
%C A116925 1 6 6 1
%C A116925 1 10 20 10 1
%C A116925 ...
%C A116925 (this is the Narayana triangle A001263)
%C A116925 and the row sums are the Catalan numbers, which give our r=2 diagonal.
%H A116925 N. J. A. Sloane, <a href="/A116925/b116925.txt">First 30 rows, flattened</a>
%F A116925 Comment from _N. J. A. Sloane_, Sep 07 2006: (Start)
%F A116925 The n-th entry in the r-th diagonal from the right (r >= 0, n >= 1) is given by the quotient:
%F A116925 Sum_{k=1..n} Product_{i=0..r-1} binomial(n+r-2, k-1+i)
%F A116925 ------------------------------------------------------
%F A116925 Product_{i=1..r-1} binomial(n+r-2, i)
%F A116925 (End)
%e A116925 The first few rows of the triangle are:
%e A116925 1
%e A116925 1 2
%e A116925 1 2 3
%e A116925 1 2 4 4
%e A116925 1 2 5 8 5
%e A116925 1 2 6 14 16 6
%e A116925 1 2 7 22 42 32 7
%e A116925 1 2 8 32 92 132 64 8
%e A116925 1 2 9 44 177 422 429 128 9
%e A116925 1 2 10 58 310 1122 2074 1430 256 10
%e A116925 ...
%p A116925 g:=proc(n,p) local k,i; 1 + add( mul( binomial(n+p-1,p+i) / binomial(n+p-1,i), i=0..k-1 ), k=1..n); end; (_N. J. A. Sloane_, based on the formula from Hsueh-Hsing Hung)
%p A116925 f:=proc(n,r) local k,b,i; b:=binomial; add( mul( b(n+r-2,k-1+i),i=0..r-1)/ mul( b(n+r-2,i),i=1..r-1),k=1..n); end; M:=30; for j from 0 to M do lprint(seq(f(i,j+1-i),i=1..j+1)); od; # _N. J. A. Sloane_
%t A116925 rows = 11; t[n_, p_] := 1 + Sum[Product[ Binomial[ n+p-1, p+i] / Binomial[ n+p-1, i], {i, 0, k-1}], {k, 1, n}]; Flatten[ Table[ t[p, n-p], {n, 0, rows}, {p, 0, n}]](* _Jean-François Alcover_, Nov 18 2011, after Maple *)
%Y A116925 Diagonals of the triangle are generalized Catalan numbers. The first few diagonals (from the right) are A000027, A000079, A000108, A001181, A005362, A005363, ... The intermediate triangles include Pascal's triangle A007318, the Narayana triangle A001263, ...
%Y A116925 Row sums give A104253.
%K A116925 nonn,tabl,nice
%O A116925 0,3
%A A116925 _Gary W. Adamson_, Feb 26 2006
%E A116925 One entry corrected by Hsueh-Hsing Hung (hhh(AT)mail.nhcue.edu.tw), Sep 06 2006
%E A116925 Edited and extended by _N. J. A. Sloane_, Sep 07 2006
%E A116925 Simpler formula provided by Hsueh-Hsing Hung (hhh(AT)mail.nhcue.edu.tw), Sep 08 2006, which is now taken as the definition of this triangle
%E A116925 Edited by _Jon E. Schoenfield_, Dec 12 2015