A207636 - OEIS

%I #14 Jan 22 2020 20:12:30

%S 1,3,2,6,7,2,12,20,11,2,24,52,42,15,2,48,128,136,72,19,2,96,304,400,

%T 280,110,23,2,192,704,1104,960,500,156,27,2,384,1600,2912,3024,1960,

%U 812,210,31,2,768,3584,7424,8960,6944,3584,1232,272,35,2,1536,7936

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A207635; see Formula section.

%C As triangle T(n,k) with 0 <= k <= n, it is (3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Feb 26 2012

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

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

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

%F From _Philippe Deléham_, Feb 26 2012: (Start)

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

%F T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0) = 1, T(1,0) = 3, T(1,1) = 2 and T(n,k) = 0 if k < 0 or if k > n.

%F G.f.: (1+x+y*x)/(1-2*x-y*x).

%F Sum_{k=0..n} T(n,k)*x^k = A003945(n), |A084244(n)|, A189274(n) for x = 0, 1, 3 respectively.

%F Sum_{k=0..n} T(n,k)*x^(n-k) = A040000(n), |A084244(n)|, A128625(n) for x = 0, 1, 2 respectively. (End)

%e First five rows:

%e    1;

%e    3,  2;

%e    6,  7,  2;

%e   12, 20, 11,  2;

%e   24, 52, 42, 15,  2;

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

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

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

%t Table[Factor[u[n, x]], {n, 1, z}]

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

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

%t TableForm[cu]

%t Flatten[%]  (* A207635 *)

%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[%]  (* A207636 *)

%Y Cf. A207635.

%Y Cf. A084938, A003945, A040000.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Feb 24 2012