A208340 Triangle of coefficients of polynomials v(n,x) jointly generated with A202390; see the Formula section.

1, 2, 2, 3, 6, 3, 4, 13, 14, 5, 5, 24, 41, 30, 8, 6, 40, 96, 109, 60, 13, 7, 62, 196, 308, 262, 116, 21, 8, 91, 364, 743, 868, 590, 218, 34, 9, 128, 630, 1604, 2413, 2240, 1267, 402, 55, 10, 174, 1032, 3186, 5926, 7046, 5424, 2627, 730, 89, 11, 230, 1617

Offset: 1

Author

Clark Kimberling, Feb 27 2012

Keywords

• nonn
• tabl

Comments

v(n,n) = F(n+1), where F=A000045, the Fibonacci numbers.

Alternating row sums of v: (1,0,0,0,0,0,0,0,...).

As triangle T(n,k) with 0 <= k <= n, it is (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Examples

			First five rows:
  1;
  2,  2;
  3,  6,  3;
  4, 13, 14,  5;
  5, 24, 41, 30,  8;
The first five polynomials v(n,x):
  1
  2 +  2x
  3 +  6x +  3x^2
  4 + 13x + 14x^2 +  5x^3
  5 + 24x + 41x^2 + 30x^3 + 8x^4
		

Crossrefs

Cf. A202390.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A202390 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208340 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* u row sums *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* v row sums *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* u alt. row sums *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* v alt. row sums *)

Formula

u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x), where u(1,x)=1, v(1,x)=1.

From Philippe Deléham, Feb 28 2012: (Start)

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

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

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

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000027(n+1), A003946(n), A109115(n), A180031(n) for x = -1, 0, 1, 2, 3 respectively. (End)