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 A210595 #10 May 25 2021 01:37:56
%S A210595 1,2,1,3,3,2,4,6,7,3,5,10,16,13,5,6,15,30,35,25,8,7,21,50,75,76,46,13,
%T A210595 8,28,77,140,181,157,84,21,9,36,112,238,371,413,317,151,34,10,45,156,
%U A210595 378,686,924,911,625,269,55,11,55,210,570,1176,1848,2206,1949,1211,475,89
%N A210595 Triangle of coefficients of polynomials v(n,x) jointly generated with A209999; see the Formula section.
%C A210595 Row n starts with n and ends with F(n), where F=A000045 (Fibonacci numbers).
%C A210595 Row sums: A048739.
%C A210595 Alternating row sums: 1,1,2,2,3,3,4,4,5,5, ...
%C A210595 For a discussion and guide to related arrays, see A208510.
%H A210595 G. C. Greubel, <a href="/A210595/b210595.txt">Rows n = 1..30 of the triangle, flattened</a>
%F A210595 u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x) + 1,
%F A210595 v(n,x) = x*u(n-1,x) + v(n-1,x) + 1,
%F A210595 where u(1,x) = 1, v(1,x) = 1.
%F A210595 T(n, k) = [x^k]( v(n,x) ), where v(n, x) = (1+x)*v(n-1, x) + x^2*v(n-2, x) + 1, v(1, x) = 1, and v(2, x) = 2 + x. - _G. C. Greubel_, May 24 2021
%e A210595 First few rows are:
%e A210595 1;
%e A210595 2, 1;
%e A210595 3, 3, 2;
%e A210595 4, 6, 7, 3;
%e A210595 5, 10, 16, 13, 5;
%e A210595 6, 15, 30, 35, 25, 8;
%e A210595 7, 21, 50, 75, 76, 46, 13;
%e A210595 First few polynomials v(n,x) are:
%e A210595 v(1, x) = 1;
%e A210595 v(2, x) = 2 + 1*x;
%e A210595 v(3, x) = 3 + 3*x + 2*x^2;
%e A210595 v(4, x) = 4 + 6*x + 7*x^2 + 3*x^3;
%e A210595 v(5, x) = 5 + 10*x + 16*x^2 + 13*x^3 + 5*x^4;
%t A210595 (* First program *)
%t A210595 u[1, x_]:= 1; v[1, x_]:= 1; z = 16;
%t A210595 u[n_, x_]:= x*u[n-1, x] + (1+x)*v[n-1, x] + 1;
%t A210595 v[n_, x_]:= x*u[n-1, x] + v[n-1, x] + 1;
%t A210595 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210595 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210595 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210595 TableForm[cu]
%t A210595 Flatten[%] (* A210565 *)
%t A210595 Table[Expand[v[n, x]], {n, 1, z}]
%t A210595 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210595 TableForm[cv]
%t A210595 Flatten[%] (* A210595 *)
%t A210595 (* Second program *)
%t A210595 v[n_, x_]:= v[n, x]= If[n<2, n+1 +n*x, (1+x)*v[n-1, x] +x^2*v[n-2, x] +1];
%t A210595 T[n_]:= CoefficientList[Series[v[n, x], {x,0,n}], x];
%t A210595 Table[T[n-1], {n, 12}]//Flatten (* _G. C. Greubel_, May 24 2021 *)
%o A210595 (Sage)
%o A210595 @CachedFunction
%o A210595 def v(n,x): return n+1+n*x if (n<2) else (1+x)*v(n-1,x) +x^2*v(n-2,x) +1
%o A210595 def T(n): return taylor( v(n,x) , x,0,n).coefficients(x, sparse=False)
%o A210595 flatten([T(n-1) for n in (1..12)]) # _G. C. Greubel_, May 24 2021
%Y A210595 Cf. A208510, A210565.
%K A210595 nonn,tabl
%O A210595 1,2
%A A210595 _Clark Kimberling_, Mar 23 2012