A208337 Triangle of coefficients of polynomials v(n,x) jointly generated with A208836; see the Formula section.

1, 1, 2, 1, 3, 3, 1, 4, 7, 5, 1, 5, 12, 15, 8, 1, 6, 18, 31, 30, 13, 1, 7, 25, 54, 73, 58, 21, 1, 8, 33, 85, 145, 162, 109, 34, 1, 9, 42, 125, 255, 361, 344, 201, 55, 1, 10, 52, 175, 413, 701, 850, 707, 365, 89, 1, 11, 63, 236, 630, 1239, 1806, 1918, 1416, 655

Offset

1,3

Comments

coef. of x(n-1) in u(n,x): A000045(n), Fibonacci numbers

coef. of x(n-1) in v(n,x): A000045(n+1)

row sums, u(n,1): A000129

row sums, v(n,1): A001333

alternating row sums, u(n,-1): 1,0,1,0,1,0,1,0,1,0,...

alternating row sums, v(n,-1): 1,-1,1,-1,1,-1,1,-1,...

Subtriangle of the triangle given by (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 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..65.

C. Kimberling, Enumeration of paths, compositions of integers and Fibonacci numbers, Fib. Quarterly 39 (5) (2001) 430-435 Figure 2.

Formula

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

v(n,x)=(x+1)*u(n-1,x)+x*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-y*x+y*x^2-y^2*x^2)/(1-x-y*x-y^2*x^2).

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

G.f.: -(1+x*y)*x*y/(-1+x*y+x^2*y^2+x). - R. J. Mathar, Aug 11 2015

Example

First five rows:
  1
  1...2
  1...3...3
  1...4...7....5
  1...5...12...15...8
First five polynomials v(n,x):
  1
  1 + 2x
  1 + 3x + 3x^2
  1 + 4x + 7x^2 + 5x^3
  1 + 5x + 12x^2 + 15x^3 + 8x^4
(1, 0, -1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, ...) begins :
  1
  1, 0
  1, 2, 0
  1, 3, 3, 0
  1, 4, 7, 5, 0
  1, 5, 12, 15, 8, 0
  1, 6, 18, 31, 30, 13, 0
  1, 7, 25, 54, 73, 58, 21, 0 . Philippe Deléham, Apr 09 2012

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*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[%]    (* A208336 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208337 *)
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*)

Crossrefs

Cf. A208336.

Sequence in context: A133804 A185943 A352001 * A208335 A208597 A179943

Adjacent sequences: A208334 A208335 A208336 * A208338 A208339 A208340

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 26 2012

Status

approved