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 A306309 #16 Feb 09 2019 11:12:45
%S A306309 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,5,1,1,5,1,1,6,6,2,6,6,1,1,7,12,8,8,
%T A306309 12,7,1,1,8,19,2,16,2,19,8,1,1,9,27,21,18,18,21,27,9,1,1,1,36,48,39,
%U A306309 36,39,48,36,1,1,1,2,37,84,87,75,75,87,84,37,2,1
%N A306309 The "zeroless Pascal triangle" read by rows.
%C A306309 Left and right edges are all 1's, interior entries are obtained by removing zeros from the sum of the two numbers above them.
%C A306309 For any k >= 0 and n >= 0, let d_k(n) = T(n+k, k).
%C A306309 For any k >= 0, d_k is eventually periodic: by induction:
%C A306309 - for k = 0: for any n >= 0, d_0(n) = 1, hence d_0 is 1-periodic,
%C A306309 - suppose that the property is true for some k >= 0,
%C A306309 - d_k is eventually p_k-periodic, and so d_k is bounded, say by m_k,
%C A306309 - d_{k+1}(n+1) - d_{k+1}(n) = d_k(n+1) <= m_k,
%C A306309 - so the first difference of d_{k+1} is bounded by m_k,
%C A306309 - A004719 has arbitrary large gaps,
%C A306309 and we can choose a range of m_k+1 terms that do not belong to A004719,
%C A306309 say x_k..x_k+m_k (with x_k > 1),
%C A306309 - d_{k+1}(0) = 1 < x_k,
%C A306309 and if d_{k+1}(n) < x_k, then d_{k+1)(n+1) < x_k,
%C A306309 so d_{k+1} is bounded by x_k,
%C A306309 - let D_{k+1}(n) = d_{k+1}(n*p_k},
%C A306309 - D_{k+1} is bounded,
%C A306309 so D_{k+1}(n + q_k) = D_{k+1}(n) for some n >= 0 and q_k > 0,
%C A306309 - we can assume that n*p_k is beyond the transient part of d_k,
%C A306309 - d_{k+1}(n*p_k + q_k*p_k + 1) = d_{k+1}(n*p_k+q_k*p_k) + d_k(n*p_k+q_k*p_k + 1)
%C A306309 = d_{k+1}(n*p_k) + d_k(n*p_k + 1)
%C A306309 = d_{k+1}(n*p_k + 1),
%C A306309 - we can generalize: for any m >= n*p_k, d_{k+1}(m + q_k*p_k) = d_{k+1)(m),
%C A306309 - and d_{k+1} is (at least q_k*p_k-)periodic, QED.
%H A306309 Rémy Sigrist, <a href="/A306309/a306309.png">Colored representation of the first 1000 rows</a> (where the hue is function of T(n, k))
%H A306309 Rémy Sigrist, <a href="/A306309/a306309.gp.txt">PARI program for A306309</a>
%F A306309 T(n, 0) = T(n, n) = 1 for n >= 0.
%F A306309 T(n, k) = A004719(T(n-1, k-1) + T(n-1, k)) for n >= 0 and k = 1..n-1.
%F A306309 T(n, 1) = A177274(n-1) for any n > 0.
%e A306309 Triangle begins:
%e A306309 1
%e A306309 1 1
%e A306309 1 2 1
%e A306309 1 3 3 1
%e A306309 1 4 6 4 1
%e A306309 1 5 1 1 5 1
%e A306309 1 6 6 2 6 6 1
%e A306309 1 7 12 8 8 12 7 1
%e A306309 1 8 19 2 16 2 19 8 1
%e A306309 1 9 27 21 18 18 21 27 9 1
%e A306309 ...
%o A306309 (PARI) See Links section.
%Y A306309 Cf. A004719, A007318, A177274.
%K A306309 nonn,tabl,base
%O A306309 0,5
%A A306309 _Rémy Sigrist_, Feb 06 2019