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

1,6

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

Links

Table of n, a(n) for n=1..75.

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

Example

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

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 *)

Crossrefs

Cf. A208343, A208510.

Sequence in context: A296313 A183342 A046688 * A157283 A067049 A349976

Adjacent sequences: A208339 A208340 A208341 * A208343 A208344 A208345

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Feb 25 2012

Status

approved