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A209745 Triangle of coefficients of polynomials u(n,x) jointly generated with A209746; see the Formula section. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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 A025498 A193693

Adjacent sequences:  A209742 A209743 A209744 * A209746 A209747 A209748

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 13 2012

STATUS

approved

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Last modified October 16 15:51 EDT 2019. Contains 328101 sequences. (Running on oeis4.)