A209414 - OEIS

%I #19 Feb 08 2024 05:28:31

%S 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 1,1,19,131,403,553,301,49,1,1,22,184,736,1462,1372,532,64,1,1,25,246,

%U 1216,3206,4446,3024,876,81,1,1,28,317,1870,6190,11584,11826,6084,1365,100,1

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

%C For a discussion and guide to related arrays, see A208510.

%C 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 G. C. Greubel, <a href="/A209414/b209414.txt">Table of n, a(n) for n = 1..1225</a>

%F u(n,x) = x*u(n-1,x) + v(n-1,x),

%F v(n,x) = 2x*u(n-1,x) + (x+1)*v(n-1,x),

%F where u(1,x)=1, v(1,x)=1.

%F From _Philippe Deléham_, Apr 01 2012: (Start)

%F As DELTA-triangle T(n,k) with 0 <= k <= n:

%F 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 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 First five rows:

%e   1;

%e   1,  1;

%e   1,  4,  1;

%e   1,  7,  9,  1;

%e   1, 10, 26, 16,  1;

%e First three polynomials v(n,x):

%e   1

%e   1 + x

%e   1 + 4x + x^2.

%e From _Philippe Deléham_, Apr 01 2012: (Start)

%e (1, 0, 2, -2, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, ...) begins:

%e   1;

%e   1,  0;

%e   1,  1,  0;

%e   1,  4,  1,  0;

%e   1,  7,  9,  1,  0;

%e   1, 10, 26, 16,  1,  0;

%e   1, 13, 52, 70, 25,  1,  0; (End)

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];

%t v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x];

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%]    (* A209414 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%]    (* A112351 *)

%t 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 Cf. A112351, A208510.

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_, Mar 09 2012