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 A209415 #28 May 28 2023 08:47:07
%S A209415 1,1,1,1,3,1,1,4,6,1,1,6,11,10,1,1,7,21,25,15,1,1,9,30,57,50,21,1,1,
%T A209415 10,45,99,133,91,28,1,1,12,58,168,275,280,154,36,1,1,13,78,250,523,
%U A209415 675,546,246,45,1,1,15,95,370,885,1433,1509,1002,375,55,1,1,16,120,505,1435,2718,3564,3135,1749,550,66,1
%N A209415 Triangle of coefficients of polynomials u(n,x) jointly generated with A209416; see the Formula section.
%C A209415 For a discussion and guide to related arrays, see A208510.
%C A209415 Subtriangle of the triangle given by (1, 0, 1, -2, 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_, Apr 02 2012
%C A209415 Up to reflection at the vertical axis, the triangle of numbers given here coincides with the triangle given in A208334, i.e., the numbers are the same just read row-wise in the opposite direction. - _Christine Bessenrodt_, Jul 21 2012
%H A209415 G. C. Greubel, <a href="/A209415/b209415.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A209415 u(n,x) = x*u(n-1,x) + v(n-1,x),
%F A209415 v(n,x) = (x+1)*u(n-1,x) + x*v(n-1,x),
%F A209415 where u(1,x)=1, v(1,x)=1.
%F A209415 From _Philippe Deléham_, Apr 02 2012: (Start)
%F A209415 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A209415 G.f.: (1 + x - 2*y*x - 2*y*x^2 + y^2*x^2)/(1 - 2*y*x - x^2 - y*x^2 + y^2*x^2).
%F A209415 T(n,k) = 2*T(n-1,k-1) + 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 A209415 First five rows:
%e A209415 1;
%e A209415 1, 1;
%e A209415 1, 3, 1;
%e A209415 1, 4, 6, 1;
%e A209415 1, 6, 11, 10, 1;
%e A209415 First three polynomials v(n,x): 1, 1 + x, 1 + 3x + x^2.
%e A209415 From _Philippe Deléham_, Apr 02 2012: (Start)
%e A209415 (1, 0, 1, -2, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, ...) begins:
%e A209415 1;
%e A209415 1, 0;
%e A209415 1, 1, 0;
%e A209415 1, 3, 1, 0;
%e A209415 1, 4, 6, 1, 0;
%e A209415 1, 6, 11, 10, 1, 0;
%e A209415 1, 7, 21, 25, 15, 1, 0;
%e A209415 1, 9, 30, 57, 50, 21, 1, 0;
%e A209415 1, 10, 45, 99, 133, 91, 28, 1, 0; (End)
%t A209415 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209415 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209415 v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x];
%t A209415 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209415 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209415 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209415 TableForm[cu]
%t A209415 Flatten[%] (* A209415 *)
%t A209415 Table[Expand[v[n, x]], {n, 1, z}]
%t A209415 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209415 TableForm[cv]
%t A209415 Flatten[%] (* A209416 *)
%t A209415 CoefficientList[CoefficientList[Series[(1 + x - 2*y*x - 2*y*x^2 + y^2*x^2)/(1 - 2*y*x - x^2 - y*x^2 + y^2*x^2), {x,0,10}, {y,0,10}], x], y] // Flatten (* _G. C. Greubel_, Jan 03 2018 *)
%Y A209415 Cf. A209416, A208510, A208334.
%K A209415 nonn,tabl
%O A209415 1,5
%A A209415 _Clark Kimberling_, Mar 09 2012