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

1,3

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

Links

Table of n, a(n) for n=1..53.

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

Example

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

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 *)

Crossrefs

Cf. A000045, A000079, A209746, A208510.

Sequence in context: A209763 A209761 A228526 * A249620 A358537 A025498

Adjacent sequences: A209742 A209743 A209744 * A209746 A209747 A209748

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 13 2012

Status

approved