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 A210042 #19 Oct 26 2024 04:21:38
%S A210042 1,3,5,2,7,6,2,9,12,8,2,11,20,20,10,2,13,30,40,30,12,2,15,42,70,70,42,
%T A210042 14,2,17,56,112,140,112,56,16,2,19,72,168,252,252,168,72,18,2,21,90,
%U A210042 240,420,504,420,240,90,20,2,23,110,330,660,924,924,660,330,110
%N A210042 Triangle of coefficients of polynomials u(n,x) jointly generated with A124927; see the Formula section.
%C A210042 Row sums: A000225
%C A210042 For a discussion and guide to related arrays, see A208510.
%C A210042 u(n,x) = u(n-1,x) + v(n-1,x) + 1,
%C A210042 v(n,x) = x*u(n-1,x) + x*v(n-1,x) + 1,
%C A210042 where u(1,x)=1, v(1,x)=1.
%C A210042 Subtriangle of the triangle given by (1, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 25 2012
%F A210042 First five rows:
%F A210042 1;
%F A210042 3,
%F A210042 5, 2;
%F A210042 7, 6, 2;
%F A210042 9, 12, 8, 2;
%F A210042 First three polynomials u(n,x): 1, 3, 5 + 2x.
%F A210042 Also, counting the top row as row 0, row n for n > 0 is as follows: 2n+1, 2C(n,2), 2C(n,3), ..., 2C(n,n).
%F A210042 From _Philippe Deléham_, Mar 25 2012: (Start)
%F A210042 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A210042 G.f.: (1-x-y*x+2*x^2)/(1-2*x-y*x+x^2+y*x^2).
%F A210042 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = 1, T(2,0) = 3, T(1,1) = T(2,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
%F A210042 G.f.: (1+x-x*y)*x*y/((-1+x)*(x+x*y-1)). - _R. J. Mathar_, Aug 12 2015
%e A210042 From _Philippe Deléham_, Mar 25 2012: (Start)
%e A210042 (1, 2, -2, 1, 0, 0, ...) DELTA (0, 0, 1, 0, 0, ...) begins:
%e A210042 1;
%e A210042 1, 0;
%e A210042 3, 0, 0;
%e A210042 5, 2, 0, 0;
%e A210042 7, 6, 2, 0, 0;
%e A210042 9, 12, 8, 2, 0, 0;
%e A210042 11, 20, 20, 10, 2, 0, 0;
%e A210042 13, 30, 40, 30, 12, 2, 0, 0; (End)
%t A210042 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210042 u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
%t A210042 v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210042 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210042 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210042 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210042 TableForm[cu]
%t A210042 Flatten[%] (* A210042 *)
%t A210042 Table[Expand[v[n, x]], {n, 1, z}]
%t A210042 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210042 TableForm[cv]
%t A210042 Flatten[%] (* A124927 *)
%t A210042 Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
%t A210042 Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
%t A210042 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A010701 *)
%t A210042 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000012 signed *)
%Y A210042 Cf. A124927, A208510.
%K A210042 nonn,tabf
%O A210042 1,2
%A A210042 _Clark Kimberling_, Mar 17 2012