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A209125 Triangle of coefficients of polynomials u(n,x) jointly generated with A164975; see the Formula section.

%I A209125 #16 Jan 24 2020 03:26:16
%S A209125 1,2,1,3,4,2,5,9,9,4,8,20,25,20,8,13,40,65,65,44,16,21,78,150,190,162,
%T A209125 96,32,34,147,331,490,521,392,208,64,55,272,697,1192,1473,1368,928,
%U A209125 448,128,89,495,1425,2745,3888,4185,3480,2160,960,256,144,890
%N A209125 Triangle of coefficients of polynomials u(n,x) jointly generated with A164975; see the Formula section.
%C A209125 Alternating row sums:  1,1,1,1,1,1,1,1,1,1,1,1,1,...
%C A209125 For a discussion and guide to related arrays, see A208510.
%C A209125 Subtriangle of the triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ....) DELTA (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, ....) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 21 2012
%F A209125 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F A209125 v(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F A209125 where u(1,x)=1, v(1,x)=1.
%F A209125 From _Philippe Deléham_, Mar 21 2012: (Start)
%F A209125 As DELTA-triangle with 0 <= k <= n:
%F A209125 G.f.: (1-2*y*x)/(1-x-2*y*x-x^2+y*x^2).
%F A209125 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0, T(2,0) = 2, T(n,k) = 0 f k < 0 or if k > n. (End)
%e A209125 First five rows:
%e A209125   1;
%e A209125   2,  1;
%e A209125   3,  4,  2;
%e A209125   5,  9,  9,  4;
%e A209125   8, 20, 25, 20,  8;
%e A209125 First three polynomials u(n,x):
%e A209125   1
%e A209125   2 + x
%e A209125   3 + 4x + 2x^2
%e A209125 From _Philippe Deléham_, Mar 21 2012: (Start)
%e A209125 (1, 1, -1, 0, 0, ...) DELTA (0, 1, 1, 0, 0, ...) begins:
%e A209125   1;
%e A209125   1,  0;
%e A209125   2,  1,  0;
%e A209125   3,  4,  2,  0;
%e A209125   5,  9,  9,  4, 0;
%e A209125   8, 20, 25, 20, 8, 0; (End)
%t A209125 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209125 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209125 v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A209125 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209125 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209125 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209125 TableForm[cu]
%t A209125 Flatten[%]    (* A209125 *)
%t A209125 Table[Expand[v[n, x]], {n, 1, z}]
%t A209125 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209125 TableForm[cv]
%t A209125 Flatten[%]    (* A164975 *)
%Y A209125 Cf. A164975, A208510.
%K A209125 nonn,tabl
%O A209125 1,2
%A A209125 _Clark Kimberling_, Mar 05 2012