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 A210793 #15 Apr 25 2024 09:13:09
%S A210793 1,2,1,3,4,2,6,10,8,3,9,24,27,16,5,18,51,74,62,30,8,27,108,189,200,
%T A210793 136,56,13,54,216,450,574,488,282,102,21,81,432,1026,1536,1571,1128,
%U A210793 569,184,34,162,837,2268,3864,4598,3967,2486,1118,328,55,243,1620
%N A210793 Triangle of coefficients of polynomials u(n,x) jointly generated with A210794; see the Formula section.
%C A210793 Row n starts with A038754(n) and ends with F(n), where F=A000045 (Fibonacci numbers).
%C A210793 Row sums: A000244 (powers of 3).
%C A210793 Alternating row sums: A000012 (1,1,1,1,1,1,1,1,1,1,1,...).
%C A210793 For a discussion and guide to related arrays, see A208510.
%C A210793 Subtriangle of the triangle given by (1, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 29 2012
%F A210793 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F A210793 v(n,x) = (x+2)*u(n-1,x) + (x-1)*v(n-1,x),
%F A210793 where u(1,x)=1, v(1,x)=1.
%F A210793 From _Philippe Deléham_, Mar 29 2012: (Start)
%F A210793 As DELTA(triangle T(n,k) with 0 <= k <= n:
%F A210793 G.f.: (1 + x - y*x^2 - 2*y*x^2 - y^2*x^2)/(1 - y*x - 3*x^2 - 2*y*x^2 - y^2*x^2).
%F A210793 T(n,k) = T(n-1,k-1) + 3*T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k <= n. (End)
%e A210793 First five rows:
%e A210793 1;
%e A210793 2, 1;
%e A210793 3, 4, 2;
%e A210793 6, 10, 8, 3;
%e A210793 9, 24, 27, 16, 5;
%e A210793 First three polynomials u(n,x):
%e A210793 1
%e A210793 2 + x
%e A210793 3 + 4x + 2x^2.
%e A210793 From _Philippe Deléham_, Mar 29 2012: (Start)
%e A210793 (1, 1, -1, -1, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins:
%e A210793 1;
%e A210793 1, 0;
%e A210793 2, 1, 0;
%e A210793 3, 4, 2, 0;
%e A210793 6, 10, 8, 3, 0;
%e A210793 9, 24, 27, 16, 5, 0;
%e A210793 18, 51, 74, 62, 30, 8, 0; (End)
%t A210793 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210793 u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
%t A210793 d[x_] := h + x; e[x_] := p + x;
%t A210793 v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
%t A210793 j = 1; c = 0; h = 2; p = -1; f = 0;
%t A210793 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210793 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210793 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210793 TableForm[cu]
%t A210793 Flatten[%] (* A210793 *)
%t A210793 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210793 TableForm[cv]
%t A210793 Flatten[%] (* A210794 *)
%t A210793 Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000244 *)
%t A210793 Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000244 *)
%t A210793 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000012 *)
%t A210793 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)
%Y A210793 Cf. A210794, A208510.
%K A210793 nonn,tabl
%O A210793 1,2
%A A210793 _Clark Kimberling_, Mar 26 2012