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 A349839 #21 Jul 31 2022 15:57:00
%S A349839 1,1,0,1,1,0,1,2,1,0,1,3,3,1,1,1,4,6,4,2,0,1,5,10,10,6,2,0,1,6,15,20,
%T A349839 16,8,2,0,1,7,21,35,36,24,10,2,1,1,8,28,56,71,60,34,12,3,0,1,9,36,84,
%U A349839 127,131,94,46,15,3,0,1,10,45,120,211,258,225,140,61,18,3,0,1,11,55,165,331,469,483,365,201,79,21,3,1
%N A349839 Triangle T(n,k) built by placing all ones on the left edge, [1,0,0,0] repeated on the right edge, and filling the body using the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).
%C A349839 This is the m=4 member in the sequence of triangles A007318, A059259, A118923, A349839, A349841 which have all ones on the left side, ones separated by m-1 zeros on the other side, and whose interiors obey Pascal's recurrence.
%C A349839 T(n,k) is the (n,n-k)-th entry of the (1/(1-x^4),x/(1-x)) Riordan array.
%C A349839 For n>0, T(n,n-1) = A008621(n-1).
%C A349839 For n>1, T(n,n-2) = A001972(n-2).
%C A349839 For n>2, T(n,n-3) = A122046(n).
%C A349839 Sums of rows give A115451.
%C A349839 Sums of antidiagonals give A349840.
%H A349839 Michael A. Allen and Kenneth Edwards, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Allen/allen3.html">On Two Families of Generalizations of Pascal's Triangle</a>, J. Int. Seq. 25 (2022) Article 22.7.1.
%F A349839 G.f.: (1-x*y)/((1-(x*y)^4)(1-x-x*y)) in the sense that T(n,k) is the coefficient of x^n*y^k in the series expansion of the g.f.
%F A349839 T(n,0) = 1.
%F A349839 T(n,n) = delta(n mod 4,0).
%F A349839 T(n,1) = n-1 for n>0.
%F A349839 T(n,2) = (n-1)*(n-2)/2 for n>1.
%F A349839 T(n,3) = (n-1)*(n-2)*(n-3)/6 for n>2.
%F A349839 T(n,4) = C(n-1,4) + 1 for n>3.
%F A349839 T(n,5) = C(n-1,5) + n - 5 for n>4.
%F A349839 For 0 <= k < n, T(n,k) = (n-k)*Sum_{j=0..floor(k/4)} binomial(n-4*j,n-k)/(n-4*j).
%F A349839 The g.f. of the n-th subdiagonal is 1/((1-x^4)(1-x)^n).
%e A349839 Triangle begins:
%e A349839 1;
%e A349839 1, 0;
%e A349839 1, 1, 0;
%e A349839 1, 2, 1, 0;
%e A349839 1, 3, 3, 1, 1;
%e A349839 1, 4, 6, 4, 2, 0;
%e A349839 1, 5, 10, 10, 6, 2, 0;
%e A349839 1, 6, 15, 20, 16, 8, 2, 0;
%e A349839 1, 7, 21, 35, 36, 24, 10, 2, 1;
%e A349839 1, 8, 28, 56, 71, 60, 34, 12, 3, 0;
%e A349839 1, 9, 36, 84, 127, 131, 94, 46, 15, 3, 0;
%e A349839 1, 10, 45, 120, 211, 258, 225, 140, 61, 18, 3, 0;
%e A349839 1, 11, 55, 165, 331, 469, 483, 365, 201, 79, 21, 3, 1;
%t A349839 Flatten[Table[CoefficientList[Series[(1-x*y)/((1-(x*y)^4)(1 - x - x*y)), {x, 0, 24}, {y, 0, 12}], {x, y}][[n+1,k+1]],{n,0,12},{k,0,n}]]
%Y A349839 Other members of sequence of triangles: A007318, A059259, A118923, A349841.
%Y A349839 Columns: A000012, A001477, A000217, A000292, A145126, A051745.
%Y A349839 Diagonals: A121262, A008621, A001972, A122046.
%K A349839 easy,nonn,tabl
%O A349839 0,8
%A A349839 _Michael A. Allen_, Dec 01 2021