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A209830 Triangle of coefficients of polynomials u(n,x) jointly generated with A209831; see the Formula section. 3
1, 1, 2, 1, 5, 5, 1, 7, 18, 13, 1, 10, 35, 59, 34, 1, 12, 61, 147, 185, 89, 1, 15, 90, 302, 558, 564, 233, 1, 17, 129, 527, 1324, 1986, 1685, 610, 1, 20, 170, 854, 2653, 5350, 6761, 4957, 1597, 1, 22, 222, 1278, 4811, 12066, 20383, 22277, 14406, 4181, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Each row begins with 1 and ends with an odd-indexed

Fibonacci number.

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 (1, 0, 1/2, -3/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938 . - Philippe Deléham, Mar 16 2012

LINKS

Table of n, a(n) for n=1..56.

FORMULA

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

v(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x),

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

As DELTA-triangle with 0<=k<=n : G.f.: (1+x-3*y*x-3*y*x^2+y^2*x^2)/(1-3*y*x-x^2-2*y*x^2+y^2*x^2). - Philippe Deléham, Mar 16 2012

As DELTA-triangle : T(n,k) = 3*T(n-1,k-1) + T(n-2,k) + 2*T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1 = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>n . - Philippe Deléham, Mar 16 2012

EXAMPLE

First five rows:

1

1...2

1...5....5

1...7....18...13

1...10...35...59...34

First three polynomials u(n,x): 1, 1 + 2x, 1 + 5x + 5x^2.

(1, 0, 1/2, -3/2, 0, 0, ...) DELTA (0, 2, 1/2, 1/2, 0, 0, ...) begins :

1

1, 0

1, 2, 0

1, 5, 5, 0

1, 7, 18, 13, 0

1, 10, 35, 59, 34, 0. -Philippe Deléham, Mar 16 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] + 2 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[%]    (* A209830 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]    (* A209831 *)

CROSSREFS

Cf. A209831, A208510.

Sequence in context: A111785 A304462 A021468 * A209695 A033282 A126350

Adjacent sequences:  A209827 A209828 A209829 * A209831 A209832 A209833

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 13 2012

STATUS

approved

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Last modified October 19 21:28 EDT 2019. Contains 328244 sequences. (Running on oeis4.)