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 A208334 #16 Jan 22 2020 20:12:58
%S A208334 1,1,1,1,3,1,1,6,4,1,1,10,11,6,1,1,15,25,21,7,1,1,21,50,57,30,9,1,1,
%T A208334 28,91,133,99,45,10,1,1,36,154,280,275,168,58,12,1,1,45,246,546,675,
%U A208334 523,250,78,13,1,1,55,375,1002,1509,1433,885,370,95,15,1,1,66,550
%N A208334 Triangle of coefficients of polynomials u(n,x) jointly generated with A208335; see the Formula section.
%C A208334 row sums, u(n,1): A000129
%C A208334 row sums, v(n,1): A001333
%C A208334 alternating row sums, u(n,-1): 1,0,-1,-2,-3,-4,-5,-6,...
%C A208334 alternating row sums, v(n,-1): 1,1,1,1,1,1,1,1,1,1,1,...
%C A208334 Subtriangle of the triangle (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 26 2012
%C A208334 Up to reflection at the vertical axis, the triangle of numbers given here coincides with the triangle given in A209415, i.e., the numbers are the same just read row-wise in the opposite direction. - _Christine Bessenrodt_, Jul 21 2012
%F A208334 u(n,x) = u(n-1,x) + x*v(n-1,x),
%F A208334 v(n,x) = (x+1)*u(n-1,x) + v(n-1,x),
%F A208334 where u(1,x)=1, v(1,x)=1.
%F A208334 From _Philippe Deléham_, Mar 26 2012: (Start)
%F A208334 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A208334 G.f.: (1-x-y^2*x^2)/(1-2*x-y*x^2+x^2-y^2*x^2).
%F A208334 T(n,k) = 2*T(n-1,k) - T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e A208334 First five rows:
%e A208334 1;
%e A208334 1, 1;
%e A208334 1, 3, 1;
%e A208334 1, 6, 4, 1;
%e A208334 1, 10, 11, 6, 1;
%e A208334 First five polynomials u(n,x):
%e A208334 1;
%e A208334 1 + x;
%e A208334 1 + 3x + x^2;
%e A208334 1 + 6x + 4x^2 + x^3;
%e A208334 1 + 10x + 11x^2 + 6x^3 + x^4;
%e A208334 From _Philippe Deléham_, Mar 26 2012: (Start)
%e A208334 (1, 0, 1, 0, 0, 0, ...) DELTA (0, 1, 0, -1, 0, 0, ...) begins:
%e A208334 1;
%e A208334 1, 0;
%e A208334 1, 1, 0;
%e A208334 1, 3, 1, 0;
%e A208334 1, 6, 4, 1, 0;
%e A208334 1, 10, 11, 6, 1, 0; (End)
%t A208334 u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t A208334 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208334 v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x];
%t A208334 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208334 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208334 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208334 TableForm[cu]
%t A208334 Flatten[%] (* A208334 *)
%t A208334 Table[Expand[v[n, x]], {n, 1, z}]
%t A208334 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208334 TableForm[cv]
%t A208334 Flatten[%] (* A208335 *)
%t A208334 Table[u[n, x] /. x -> 1, {n, 1, z}] (* u row sums *)
%t A208334 Table[v[n, x] /. x -> 1, {n, 1, z}] (* v row sums *)
%t A208334 Table[u[n, x] /. x -> -1, {n, 1, z}](* u alt. row sums *)
%t A208334 Table[v[n, x] /. x -> -1, {n, 1, z}](* v alt. row sums *)
%Y A208334 Cf. A208335, A209415.
%K A208334 nonn,tabl
%O A208334 1,5
%A A208334 _Clark Kimberling_, Feb 26 2012