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 A335183 #29 Aug 03 2025 23:39:41
%S A335183 0,4,4,36,60,24,288,688,560,160,2240,7080,8760,5040,1120,17304,68712,
%T A335183 114576,99456,44352,8064,133672,642824,1351840,1572480,1055040,384384,
%U A335183 59136,1034880,5864640,14912064,21778560,19536000,10695168,3294720,439296
%N A335183 T(n,k) = Sum_{j=1..n} 2^j*binomial(2*n-2*j, n-j)*binomial(n+j, n)*binomial(j, k), triangle read by rows (n >= 0 and 0 <= k <= n).
%C A335183 This was the original version of A126936.
%F A335183 T(n,n) = A069722(n+1) for n >= 0.
%F A335183 T(n,k) = A126936(n,k) = A067001(n,n-k) for n >= k >= 1.
%F A335183 T(n,0) = A126936(n,0) - binomial(2*n, n) = A067001(n,n) - A000984(n) for n >= 0.
%F A335183 Bivariate o.g.f.: Sum_{n,k >= 0} T(n,k)*x^n*y^k = -1/sqrt(1 - 4*x) + sqrt((1 + y)/(1 - 8*x*(1 + y))/(y + sqrt(1 - 8*x*(1 + y)))).
%e A335183 Table T(n,k) (with rows n >= 0 and columns k = 0..n) begins as follows:
%e A335183 0;
%e A335183 4, 4;
%e A335183 36, 60, 24;
%e A335183 288, 688, 560, 160;
%e A335183 2240, 7080, 8760, 5040, 1120;
%e A335183 17304, 68712, 114576, 99456, 44352, 8064;
%e A335183 133672, 642824, 1351840, 1572480, 1055040, 384384, 59136;
%e A335183 ...
%t A335183 t[l_, m_] := Sum[2^k*Binomial[2*m-2*k, m-k]*Binomial[m+k, m]*Binomial[k, l], {k, 1, m}]; Table[t[l, m], {m, 0, 7}, {l, 0, m}] // Flatten (* _Jean-François Alcover_, Jan 09 2014 from the original version of A126936 *)
%o A335183 (PARI) T(n,k) = sum(j=1, n, 2^j*binomial(2*n-2*j, n-j)*binomial(n+j, n)*binomial(j, k));
%o A335183 tabl(nn) = {for (n=0, nn, for (k=0, n, print1(T(n,k), ", "); ); print(); ); }
%Y A335183 Cf. A000984, A067001, A069722 (main diagonal), A126936.
%K A335183 nonn,tabl
%O A335183 0,2
%A A335183 _Petros Hadjicostas_, May 25 2020