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 A209757 #17 Jan 26 2020 17:46:16
%S A209757 1,3,2,5,8,4,7,18,20,8,9,32,56,48,16,11,50,120,160,112,32,13,72,220,
%T A209757 400,432,256,64,15,98,364,840,1232,1120,576,128,17,128,560,1568,2912,
%U A209757 3584,2816,1280,256,19,162,816,2688,6048,9408,9984,6912,2816,512
%N A209757 Triangle of coefficients of polynomials v(n,x) jointly generated with A013609; see the Formula section.
%C A209757 For a discussion and guide to related arrays, see A208510.
%C A209757 Subtriangle of the triangle given by (1, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 24 2012
%F A209757 u(n,x) = x*u(n-1,x) + x*v(n-1,x) + 1,
%F A209757 v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x) + 1,
%F A209757 where u(1,x)=1, v(1,x)=1.
%F A209757 From _Philippe Deléham_, Mar 24 2012: (Start)
%F A209757 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A209757 G.f.: (1 - x - 2*y*x + 2*x^2 + 2*x^2*y)/(1 - 2*x - 2*y*x + x^2 + 2*y*x^2).
%F A209757 T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-2), T(0,0) = T(1,0) = 1, T(1,1) = T(2,2) = 0, T(2,0) = 3, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k > n.
%F A209757 T(n,k) = 2^k*binomial(n-1,k)*(2*n-k-1)/(k+1). (End)
%F A209757 From _Peter Bala_, Dec 21 2014: (Start)
%F A209757 Following remarks assume an offset of 0.
%F A209757 T(n,k) = 2^k * A110813(n,k).
%F A209757 Riordan array ((1+x)/(1-x)^2, 2*x/(1-x)).
%F A209757 exp(2*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(2*x)*(7 + 18*x + 20*x^2/2! + 8*x^3/3!) = 7 + 32*x + 120*x^2/2! + 400*x^3/3! + 1232*x^4/4! + .... The same property holds more generally for Riordan arrays of the form (f(x), 2*x/(1-x)). (End)
%e A209757 First five rows:
%e A209757 1;
%e A209757 3, 2;
%e A209757 5, 8, 4;
%e A209757 7, 18, 20, 8;
%e A209757 9, 32, 56, 48, 16;
%e A209757 First three polynomials v(n,x):
%e A209757 1
%e A209757 3 + 2x
%e A209757 5 + 8x + 4x^2.
%e A209757 From _Philippe Deléham_, Mar 24 2012: (Start)
%e A209757 (1, 2, -2, 1, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) begins:
%e A209757 1;
%e A209757 1, 0;
%e A209757 3, 2, 0;
%e A209757 5, 8, 4, 0;
%e A209757 7, 18, 20, 8, 0;
%e A209757 9, 32, 56, 48, 16, 0; (End)
%t A209757 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209757 u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A209757 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A209757 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209757 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209757 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209757 TableForm[cu]
%t A209757 Flatten[%] (* A013609 *)
%t A209757 Table[Expand[v[n, x]], {n, 1, z}]
%t A209757 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209757 TableForm[cv]
%t A209757 Flatten[%] (* A209757 *)
%Y A209757 Cf. A013609, A208510, A110813.
%K A209757 nonn,tabl
%O A209757 1,2
%A A209757 _Clark Kimberling_, Mar 23 2012