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 A210557 #22 Apr 25 2024 09:11:55
%S A210557 1,1,2,1,3,5,1,4,10,12,1,5,16,30,29,1,6,23,56,87,70,1,7,31,91,185,245,
%T A210557 169,1,8,40,136,334,584,676,408,1,9,50,192,546,1158,1784,1836,985,1,
%U A210557 10,61,260,834,2052,3850,5312,4925,2378,1,11,73,341,1212,3366
%N A210557 Triangle of coefficients of polynomials u(n,x) jointly generated with A210558; see the Formula section.
%C A210557 Row sums: powers of 3 (see A000244).
%C A210557 For a discussion and guide to related arrays, see A208510.
%C A210557 Subtriangle of (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 23 2012
%C A210557 Up to reflection at the vertical axis, this triangle coincides with the triangle given in A164981, i.e., the numbers are the same just read row-wise in the opposite direction. - _Christine Bessenrodt_, Jul 20 2012
%F A210557 u(n,x) = x*u(n-1,x) + x*v(n-1,x)+1,
%F A210557 v(n,x) = 2x*u(n-1,x) + (x+1)v(n-1,x)+1,
%F A210557 where u(1,x)=1, v(1,x)=1.
%F A210557 From _Philippe Deléham_, Mar 23 2012. (Start)
%F A210557 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A210557 G.f.: (1 - 2*y*x + y*x^2 - y^2*x^2)/(1 - x - 2*y*x + y*x^2 - y^2*x^2).
%F A210557 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e A210557 First five rows:
%e A210557 1;
%e A210557 1, 2;
%e A210557 1, 3, 5;
%e A210557 1, 4, 10, 12;
%e A210557 1, 5, 16, 30, 29;
%e A210557 First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 5x^2.
%e A210557 From _Philippe Deléham_, Mar 23 2012: (Start)
%e A210557 (1, 0, -1/2, 1/2, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, ...) begins:
%e A210557 1;
%e A210557 1, 0;
%e A210557 1, 2, 0;
%e A210557 1, 3, 5, 0;
%e A210557 1, 4, 10, 12, 0;
%e A210557 1, 5, 16, 30, 29, 0; (End)
%t A210557 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210557 u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210557 v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A210557 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210557 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210557 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210557 TableForm[cu]
%t A210557 Flatten[%] (* A210557 *)
%t A210557 Table[Expand[v[n, x]], {n, 1, z}]
%t A210557 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210557 TableForm[cv]
%t A210557 Flatten[%] (* A210558 *)
%Y A210557 Cf. A210558, A208510, A164981.
%K A210557 nonn,tabl
%O A210557 1,3
%A A210557 _Clark Kimberling_, Mar 22 2012