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

Author

Clark Kimberling, Feb 26 2012

Keywords

• nonn
• tabl

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

Examples

			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
		

Links

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

Crossrefs

Cf. A208336.

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 + 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*)

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