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A210789 Triangle of coefficients of polynomials u(n,x) jointly generated with A210790; see the Formula section. 3
1, 1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 4, 8, 8, 5, 1, 5, 12, 18, 15, 8, 1, 6, 18, 32, 39, 28, 13, 1, 7, 24, 53, 77, 80, 51, 21, 1, 8, 32, 80, 142, 176, 160, 92, 34, 1, 9, 40, 116, 234, 352, 384, 312, 164, 55, 1, 10, 50, 160, 370, 632, 830, 812, 598, 290, 89, 1, 11, 60, 215 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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

Column 2: 1,2,3,4,5,6,7,8,...

Row sums: A006138.

Alternating row sums: signed Fibonacci numbers.

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

Subtriangle of the triangle given by (1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 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..70.

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*x^2-y^2*x^2)/(1-y*x-y*x^2-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) = T(2,1) = 1, 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, 1;

  1, 2, 2;

  1, 3, 4, 3;

  1, 4, 8, 8, 5;

First three polynomials u(n,x):

  1

  1 + x

  1 + 2x + 2x^2.

From Philippe Deléham, Mar 28 2012: (Start)

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

  1;

  1, 0;

  1, 1, 0;

  1, 2, 2, 0;

  1, 3, 4, 3, 0;

  1, 4, 8, 8, 5, 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. A210790, A208510.

Sequence in context: A179901 A209561 A283822 * A105809 A091594 A118032

Adjacent sequences:  A210786 A210787 A210788 * A210790 A210791 A210792

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 26 2012

STATUS

approved

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Last modified May 17 10:16 EDT 2022. Contains 353745 sequences. (Running on oeis4.)