A209745 Triangle of coefficients of polynomials u(n,x) jointly generated with A209746; see the Formula section.

1, 1, 2, 2, 5, 4, 3, 12, 16, 8, 5, 25, 49, 44, 16, 8, 50, 127, 166, 112, 32, 13, 96, 301, 513, 504, 272, 64, 21, 180, 670, 1408, 1808, 1424, 640, 128, 34, 331, 1427, 3562, 5641, 5816, 3824, 1472, 256, 55, 600, 2939, 8494, 15981, 20330, 17520, 9888

Offset: 1

Author

Clark Kimberling, Mar 13 2012

Keywords

• nonn
• tabl

Comments

Row n begins with F(n) and ends with 2^(n-1), where F=A000045 (Fibonacci numbers)

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

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

Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ....) DELTA (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 24 2012

Riordan array (1/(1-x-x^2), (2*x+x^2)/(1-x-x^2)). - Philippe Deléham, Mar 24 2012

Examples

			First five rows:
1
1...2
2...5....4
3...12...16...8
5...25...49...44...16
First three polynomials u(n,x): 1, 1 + 2x, 2 + 5x + 4x^2.
(0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 1, 0, 0, 0, ...) begins :
1
0, 1
0, 1, 2
0, 2, 5, 4
0, 3, 12, 16, 8
0, 5, 25, 49, 44, 16 ... - _Philippe Deléham_, Mar 24 2012
		

Crossrefs

Cf. A000045, A000079, A209746, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*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[%]    (* A209745 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209746 *)

Formula

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

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

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

T(n,k) = T(n-1, k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(1,0) = T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 24 2012