A208342 Triangle of coefficients of polynomials u(n,x) jointly generated with A208343; see the Formula section.

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 5, 1, 1, 5, 7, 10, 8, 1, 1, 6, 9, 16, 18, 13, 1, 1, 7, 11, 23, 31, 33, 21, 1, 1, 8, 13, 31, 47, 62, 59, 34, 1, 1, 9, 15, 40, 66, 101, 119, 105, 55, 1, 1, 10, 17, 50, 88, 151, 205, 227, 185, 89, 1, 1, 11, 19, 61, 113, 213, 321, 414

Offset: 1

Author

Clark Kimberling, Feb 25 2012

Keywords

• nonn
• tabl
• easy

Comments

Coefficient of x^(n-1): A000045(n) (Fibonacci numbers).

n-th row sum: 2^(n-1).

Mirror image of triangle in A053538. - Philippe Deléham, Mar 05 2012

Subtriangle of the triangle T(n,k) given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 12 2012

Examples

			First five rows:
  1
  1, 1
  1, 1, 2
  1, 1, 3, 3
  1, 1, 4, 5, 5
First five polynomials u(n,x): 1, 1 + x, 1 + x + x^2, 1 + x + 3*x^2 + 3*x^3, 1 + x + 4*x^2 + 5*x^3 + 5*x^4.
(1, 0, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
1
1, 0
1, 1, 0
1, 1, 2,  0
1, 1, 3,  3,  0
1, 1, 4,  5,  5,  0
1, 1, 5,  7, 10,  8,  0
1, 1, 6,  9, 16, 18, 13,  0
1, 1, 7, 11, 23, 31, 33, 21, 0
		

Crossrefs

Cf. A208343, 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] + 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[%]  (* A208342 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208343 *)

Formula

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

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

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

T(n,k) = A208747(n,k)/2^k. - Philippe Deléham, Mar 05 2012

From Philippe Deléham, Mar 12 2012: (Start)

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

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

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

O.g.f.: 1/(1 - z - x*z(1 - z + x*z)) = 1 + (1 + x)*z + (1 + x + 2*x^2)*z^2 + (1 + x + 3*x + 3*x^2)*z^3 + .... - Peter Bala, Dec 31 2015

u(n,x) = Sum_{j=1..floor((n+1)/2)} (-1)^(j-1)*binomial(n-j,j-1)*(x*(1-x))^(j-1)* (1+x)^(n+1-2*j) for n>=1. - Werner Schulte, Mar 07 2017

T(n,k) = Sum_{j=0..floor((k-1)/2)} binomial(k-1-j,j)*binomial(n-k+j,j) for k,n>0 and k<=n (conjectured). - Werner Schulte, Mar 07 2017