A209127 Triangle of coefficients of polynomials v(n,x) jointly generated with A209126; see the Formula section.

1, 0, 2, 0, 2, 3, 0, 2, 5, 5, 0, 2, 7, 12, 8, 0, 2, 9, 21, 25, 13, 0, 2, 11, 32, 53, 50, 21, 0, 2, 13, 45, 94, 124, 96, 34, 0, 2, 15, 60, 150, 250, 273, 180, 55, 0, 2, 17, 77, 223, 445, 617, 577, 331, 89, 0, 2, 19, 96, 315, 728, 1212, 1444, 1181, 600, 144, 0, 2, 21

Offset: 1

Author

Clark Kimberling, Mar 05 2012

Keywords

• nonn
• tabl

Comments

u(n,n)=(1,2,3,5,8,13,21,...)=A000045(n+1), Fibonacci numbers.

Alternating row sums: (1,-2,1,-2,1,-2,1,-2,...

For a discussion and guide to related arrays, see A208510.

As triangle T(n,k) with 0<=k<=n, it is (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012

Examples

			First five rows:
1
0...2
0...2...3
0...2...5...5
0...2...7...12...8
First three polynomials v(n,x): 1, 2x, 2x + 3x^2.
		

Crossrefs

Cf. A209126, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*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[%]    (* A209126 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209127 *)

Formula

u(n,x)=u(n-1,x)+(x+1)*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) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 21 2012

G.f.: (-1-x*y+x)*x*y/(-1+x*y+x+x^2*y^2). - R. J. Mathar, Aug 12 2015