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A209144 Triangle of coefficients of polynomials v(n,x) jointly generated with A209143; see the Formula section.

%I A209144 #16 Jan 22 2020 02:42:23
%S A209144 1,3,6,1,12,5,24,16,1,48,44,7,96,112,30,1,192,272,104,9,384,640,320,
%T A209144 48,1,768,1472,912,200,11,1536,3328,2464,720,70,1,3072,7424,6400,2352,
%U A209144 340,13,6144,16384,16128,7168,1400,96,1,12288,35840,39680,20736
%N A209144 Triangle of coefficients of polynomials v(n,x) jointly generated with A209143; see the Formula section.
%C A209144 Alternating row sums: 1,3,5,7,9,11,13,15,17,...
%C A209144 For a discussion and guide to related arrays, see A208510.
%C A209144 Subtriangle of the triangle given by (3,-1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 07 2012
%F A209144 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F A209144 v(n,x) = u(n-1,x) + v(n-1,x) + 1,
%F A209144 where u(1,x)=1, v(1,x)=1.
%F A209144 From _Philippe Deléham_, Mar 07 2012: (Start)
%F A209144 As triangle T(n,k) with 0 <= k <= n:
%F A209144 T(n,k) = 2*T(n-1,k) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 0 and T(n,k) = 0 if k < 0 or if k > n.
%F A209144 G.f.: (1+x)/(1-2*x-y*x^2).
%F A209144 Sum_{k=0..n} T(n,k)*x^k = A005408(n), A003945(n), A078057(n), A028859(n), A000244(n), A063782(n), A180168(n) for x = -1, 0, 1, 2, 3, 4, 5 respectively. (End)
%e A209144 First five rows:
%e A209144    1;
%e A209144    3;
%e A209144    6,  1;
%e A209144   12,  5;
%e A209144   24, 16, 1;
%e A209144 First three polynomials v(n,x): 1, 3, 6 + x.
%e A209144 (3,-1, 0, 0, 0, ...) DELTA (0, 1/3, -1/3, 0, 0, ...) begins:
%e A209144     1;
%e A209144     3,   0;
%e A209144     6,   1,   0;
%e A209144    12,   5,   0, 0;
%e A209144    24,  16,   1, 0, 0;
%e A209144    48,  44,   7, 0, 0, 0;
%e A209144    96, 112,  30, 1, 0, 0, 0;
%e A209144   192, 272, 104, 9, 0, 0, 0, 0;
%t A209144 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209144 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209144 v[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
%t A209144 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209144 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209144 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209144 TableForm[cu]
%t A209144 Flatten[%]    (* A209143 *)
%t A209144 Table[Expand[v[n, x]], {n, 1, z}]
%t A209144 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209144 TableForm[cv]
%t A209144 Flatten[%]    (* A209144 *)
%Y A209144 Cf. A209143, A208510.
%K A209144 nonn,tabf
%O A209144 1,2
%A A209144 _Clark Kimberling_, Mar 06 2012