A209705 Triangle of coefficients of polynomials u(n,x) jointly generated with A209706; see the Formula section.

1, 0, 2, 0, 3, 4, 0, 4, 10, 8, 0, 5, 18, 28, 16, 0, 6, 28, 64, 72, 32, 0, 7, 40, 120, 200, 176, 64, 0, 8, 54, 200, 440, 576, 416, 128, 0, 9, 70, 308, 840, 1456, 1568, 960, 256, 0, 10, 88, 448, 1456, 3136, 4480, 4096, 2176, 512, 0, 11, 108, 624, 2352, 6048

Offset

1,3

Comments

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

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

Links

Table of n, a(n) for n=1..61.

Formula

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

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

where u(1,x)=1, v(1,x)=1.

T(n,k) = 2*T(n-1,k)+2*T(n-1,k-1)-T(n-2,k)-2*T(n-2,k-1), T(1,0)=1, T(2,0)=0, T(2,1)=2, T(3,0)=0, T(3,1)=3, T(3,2)=4, T(n,k)=0 if k<0 or if k>=n. - Philippe Deléham, Dec 27 2013

Example

First five rows:
1
0...2
0...3...4
0...4...10...8
0...5...18...28...16
First three polynomials v(n,x): 1, 2x, 3x + 4x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
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[%]  (* A209705 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A209706 *)

Crossrefs

Cf. A209706, A208510.

Sequence in context: A227595 A078436 A368090 * A181289 A229032 A352835

Adjacent sequences: A209702 A209703 A209704 * A209706 A209707 A209708

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 12 2012

Status

approved