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A209414 Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.

%I A209414 #19 Feb 08 2024 05:28:31
%S A209414 1,1,1,1,4,1,1,7,9,1,1,10,26,16,1,1,13,52,70,25,1,1,16,87,190,155,36,
%T A209414 1,1,19,131,403,553,301,49,1,1,22,184,736,1462,1372,532,64,1,1,25,246,
%U A209414 1216,3206,4446,3024,876,81,1,1,28,317,1870,6190,11584,11826,6084,1365,100,1
%N A209414 Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.
%C A209414 For a discussion and guide to related arrays, see A208510.
%C A209414 Subtriangle of the triangle given by (1, 0, 2, -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 01 2012
%H A209414 G. C. Greubel, <a href="/A209414/b209414.txt">Table of n, a(n) for n = 1..1225</a>
%F A209414 u(n,x) = x*u(n-1,x) + v(n-1,x),
%F A209414 v(n,x) = 2x*u(n-1,x) + (x+1)*v(n-1,x),
%F A209414 where u(1,x)=1, v(1,x)=1.
%F A209414 From _Philippe Deléham_, Apr 01 2012: (Start)
%F A209414 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A209414 G.f.: (1-2*y*x-2*y*x^2+y^2*x^2)/(1-x-2*y*x-y*x^2+y^2*x^2).
%F A209414 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + 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 A209414 First five rows:
%e A209414   1;
%e A209414   1,  1;
%e A209414   1,  4,  1;
%e A209414   1,  7,  9,  1;
%e A209414   1, 10, 26, 16,  1;
%e A209414 First three polynomials v(n,x):
%e A209414   1
%e A209414   1 + x
%e A209414   1 + 4x + x^2.
%e A209414 From _Philippe Deléham_, Apr 01 2012: (Start)
%e A209414 (1, 0, 2, -2, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, ...) begins:
%e A209414   1;
%e A209414   1,  0;
%e A209414   1,  1,  0;
%e A209414   1,  4,  1,  0;
%e A209414   1,  7,  9,  1,  0;
%e A209414   1, 10, 26, 16,  1,  0;
%e A209414   1, 13, 52, 70, 25,  1,  0; (End)
%t A209414 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209414 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209414 v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209414 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209414 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209414 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209414 TableForm[cu]
%t A209414 Flatten[%]    (* A209414 *)
%t A209414 Table[Expand[v[n, x]], {n, 1, z}]
%t A209414 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209414 TableForm[cv]
%t A209414 Flatten[%]    (* A112351 *)
%t A209414 CoefficientList[CoefficientList[Series[(1 - 2*y*x - 2*y*x^2 + y^2*x^2)/(1 - x - 2*y*x - y*x^2 + y^2*x^2), {x,0,10}, {y,0,10}], x], y] // Flatten (* _G. C. Greubel_, Jan 03 2018 *)
%Y A209414 Cf. A112351, A208510.
%K A209414 nonn,tabl
%O A209414 1,5
%A A209414 _Clark Kimberling_, Mar 09 2012