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

1,6

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

Links

Table of n, a(n) for n=1..69.

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

Example

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)

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}]

Crossrefs

Cf. A208345, A208510.

Sequence in context: A105687 A209415 A058879 * A209172 A263950 A160870

Adjacent sequences: A208341 A208342 A208343 * A208345 A208346 A208347

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 25 2012

Extensions

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

Status

approved