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A209139 Triangle of coefficients of polynomials u(n,x) jointly generated with A209140; see the Formula section. 3
1, 2, 1, 3, 5, 3, 5, 12, 15, 7, 8, 27, 45, 42, 17, 13, 55, 119, 151, 116, 41, 21, 108, 282, 458, 480, 315, 99, 34, 205, 630, 1228, 1631, 1467, 845, 239, 55, 381, 1343, 3054, 4849, 5502, 4358, 2244, 577, 89, 696, 2769, 7173, 13218, 17895, 17838, 12666 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Column 1: A000045 (Fibonacci numbers).

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

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

Subtriangle of the triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 11 2012

LINKS

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

FORMULA

u(n,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.

From Philippe Deléham, Apr 11 2012: (Start)

As DELTA-triangle T(n,k) with 0 <= k <= n:

G.f.: (1-2*y*x-y*x^2-y^2*x^2)/(1-x-x^2-2*y*x-y^2*x^2).

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

EXAMPLE

First five rows:

1;

2, 1;

3, 5, 3;

5, 12, 15, 7;

8, 27, 45, 42, 17;

First three polynomials u(n,x):

1

2 + x

3 + 5x + 3x^2

From Philippe Deléham, Apr 11 2012: (Start)

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

1;

1, 0;

2, 1, 0;

3, 5, 3, 0;

5, 12, 15, 7, 0;

8, 27, 45, 42, 17, 0;

13, 55, 119, 151, 116, 41, 0; (End)

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, 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[%] (* A209139 *)

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

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

TableForm[cv]

Flatten[%] (* A209140 *)

CROSSREFS

Cf. A209140, A208510.

Sequence in context: A258236 A258244 A258248 * A257161 A253676 A182939

Adjacent sequences: A209136 A209137 A209138 * A209140 A209141 A209142

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 05 2012

STATUS

approved

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Last modified December 4 20:32 EST 2022. Contains 358570 sequences. (Running on oeis4.)