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A209131 Triangle of coefficients of polynomials u(n,x) jointly generated with A209132; see the Formula section.

%I A209131 #13 Jan 24 2020 03:26:35
%S A209131 1,2,1,2,4,3,2,8,12,5,2,12,28,28,11,2,16,52,84,68,21,2,20,84,188,236,
%T A209131 156,43,2,24,124,356,612,628,356,85,2,28,172,604,1324,1852,1612,796,
%U A209131 171,2,32,228,948,2532,4500,5316,4020,1764,341,2,36,292,1404
%N A209131 Triangle of coefficients of polynomials u(n,x) jointly generated with A209132; see the Formula section.
%C A209131 For a discussion and guide to related arrays, see A208510.
%C A209131 Subtriangle of the triangle given by (1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 21 2012
%F A209131 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F A209131 v(n,x) = 2x*u(n-1,x) + x*v(n-1,x),
%F A209131 where u(1,x)=1, v(1,x)=1.
%F A209131 From _Philippe Deléham_, Mar 21 2012: (Start)
%F A209131 As DELTA-triangle with 0 <= k <= n:
%F A209131 G.f.: (1-y*x+x^2-y*x^2-2*y^2*x^2)/(1-x-y*x-y*x^2-2*y^2*x^2).
%F A209131 T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + 2*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 A209131 First five rows:
%e A209131   1;
%e A209131   2,  1;
%e A209131   2,  4,  3;
%e A209131   2,  8, 12,  5;
%e A209131   2, 12, 28, 28, 11;
%e A209131 First three polynomials u(n,x):
%e A209131   1
%e A209131   2 + x
%e A209131   2 + 4x + 3x^2
%e A209131 From _Philippe Deléham_, Mar 21 2012: (Start)
%e A209131 (1, 1, -2, 1, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, ...) begins:
%e A209131   1;
%e A209131   1,   0;
%e A209131   2,   1,   0;
%e A209131   2,   4,   3,   0;
%e A209131   2,   8,  12,   5,   0;
%e A209131   2,  12,  28,  28,  11,   0;
%e A209131   2,  16,  52,  84,  68,  21,   0;
%e A209131   2,  20,  84, 188, 236, 156,  43,   0; (End)
%t A209131 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209131 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209131 v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x];
%t A209131 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209131 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209131 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209131 TableForm[cu]
%t A209131 Flatten[%]    (* A209131 *)
%t A209131 Table[Expand[v[n, x]], {n, 1, z}]
%t A209131 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209131 TableForm[cv]
%t A209131 Flatten[%]    (* A209132 *)
%Y A209131 Cf. A209132, A208510.
%K A209131 nonn,tabl
%O A209131 1,2
%A A209131 _Clark Kimberling_, Mar 05 2012