A208344 Triangle of coefficients of polynomials u(n,x) jointly generated with A208345; see the Formula section.

1, 1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 5, 10, 17, 1, 1, 6, 13, 27, 41, 1, 1, 7, 16, 38, 71, 99, 1, 1, 8, 19, 50, 106, 186, 239, 1, 1, 9, 22, 63, 146, 294, 484, 577, 1, 1, 10, 25, 77, 191, 424, 806, 1253, 1393, 1, 1, 11, 28, 92, 241, 577, 1212, 2191, 3229, 3363, 1, 1, 12

Offset: 1

Author

Clark Kimberling, Feb 25 2012

Keywords

• nonn
• tabl

Comments

Row sums, u(n,1): (1,2,5,13,...), odd-indexed Fibonacci numbers.

Row sums, v(n,1): (1,3,8,21,...), even-indexed Fibonacci numbers.

Subtriangle of the triangle given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 09 2012

Examples

			First five rows:
  1;
  1,  1;
  1,  1,  3;
  1,  1,  4,  7;
  1,  1,  5, 10, 17;
First five polynomials u(n,x):
  1
  1 + x
  1 + x + 3x^2
  1 + x + 4x^2 + 7x^3
  1 + x + 5x^2 + 10x^3 + 17x^4.
From _Philippe Deléham_, Apr 09 2012: (Start)
(1, 0, -1, 1, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, ...) begins:
  1;
  1,  0;
  1,  1,  0;
  1,  1,  3,  0;
  1,  1,  4,  7,  0;
  1,  1,  5, 10, 17,  0;
  1,  1,  6, 13, 27, 41,  0; (End)
		

Crossrefs

Cf. A208345, A208510.

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*u[n - 1, x] + 2 x*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[%]    (* A208344 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208345 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]
Table[v[n, x] /. x -> 1, {n, 1, z}]

Formula

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

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

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

From Philippe Deléham, Apr 09 2012: (Start)

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

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

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) -2*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)

Working with an offset of 0, the row reversed triangle is the Riordan array ( (1 - x)/(1 - 2*x - x^2), x*(1 - 2*x)/(1 - 2*x - x^2) ) with g.f. (1 - x)/(1 - (2 + y)*x - (1 - 2*y)*x^2) = 1 + (1 + y)*x + (3 + y + y^2)*x^2 + (7 + 4*y + y^2 + y^3)*x^3 + .... - Peter Bala, Jun 01 2024

Extensions

a(69) corrected by Georg Fischer, Sep 03 2021