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

1,3

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

Links

Table of n, a(n) for n=1..69.

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

Example

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.

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 *)

Crossrefs

Cf. A209126, A208510.

Sequence in context: A067871 A198632 A060155 * A127954 A198061 A265583

Adjacent sequences: A209124 A209125 A209126 * A209128 A209129 A209130

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 05 2012

Status

approved