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A210790 Triangle of coefficients of polynomials v(n,x) jointly generated with A210789; see the Formula section. 3
1, 1, 2, 1, 2, 3, 1, 4, 5, 5, 1, 4, 10, 10, 8, 1, 6, 14, 24, 20, 13, 1, 6, 21, 38, 52, 38, 21, 1, 8, 27, 65, 96, 109, 71, 34, 1, 8, 36, 92, 176, 224, 220, 130, 55, 1, 10, 44, 136, 280, 446, 500, 434, 235, 89, 1, 10, 55, 180, 440, 772, 1066, 1074, 839, 420, 144, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row n starts with 1 and ends with F(n+1), where F=A000045 (Fibonacci numbers).

Column 2: 2,2,4,4,6,6,8,8,...

Row sums: A105476.

Alternating row sums: signed Fibonacci numbers.

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

Subtriangle of the triangle given by (1, 0, -1, 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 28 2012

LINKS

Table of n, a(n) for n=1..67.

FORMULA

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

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

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

From Philippe Deléham, Mar 28 2012: (Start)

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

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

T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 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;

  1,  2;

  1,  2,  3;

  1,  4,  5,  5;

  1,  4, 10, 10,  8;

First three polynomials v(n,x):

  1

  1 + 2x

  1 + 2x + 3x^2.

From Philippe Deléham, Mar 28 2012: (Start)

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

  1;

  1,  0;

  1,  2,  0;

  1,  2,  3,  0;

  1,  4,  5,  5,  0;

  1,  4, 10, 10,  8,  0; (End)

MATHEMATICA

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

u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;

d[x_] := h + x; e[x_] := p + x;

v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;

j = 0; c = 0; h = 2; p = -1; f = 0;

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[%]    (* A210789 *)

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

TableForm[cv]

Flatten[%]    (* A210790 *)

Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A006138 *)

Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A105476 *)

Table[u[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)

Table[v[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)

CROSSREFS

Cf. A210789, A208510.

Sequence in context: A055884 A055889 A125930 * A224653 A101391 A327632

Adjacent sequences:  A210787 A210788 A210789 * A210791 A210792 A210793

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 26 2012

STATUS

approved

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Last modified May 20 06:37 EDT 2022. Contains 353852 sequences. (Running on oeis4.)