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A210793 Triangle of coefficients of polynomials u(n,x) jointly generated with A210794; see the Formula section. 3
1, 2, 1, 3, 4, 2, 6, 10, 8, 3, 9, 24, 27, 16, 5, 18, 51, 74, 62, 30, 8, 27, 108, 189, 200, 136, 56, 13, 54, 216, 450, 574, 488, 282, 102, 21, 81, 432, 1026, 1536, 1571, 1128, 569, 184, 34, 162, 837, 2268, 3864, 4598, 3967, 2486, 1118, 328, 55, 243, 1620 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row n starts with A038754(n) and ends with F(n), where F=A000045 (Fibonacci numbers).

Row sums: A000244 (powers of 3).

Alternating row sums: A000012 (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, -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 29 2012

LINKS

Table of n, a(n) for n=1..57.

FORMULA

u(n,x) = u(n-1,x) + (x+1)*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 29 2012: (Start)

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

G.f.: (1 + x - y*x^2 - 2*y*x^2 - y^2*x^2)/(1 - y*x - 3*x^2 - 2*y*x^2 - y^2*x^2).

T(n,k) = T(n-1,k-1) + 3*T(n-2,k) + 2*T(n-2,k-1) + 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.

EXAMPLE

First five rows:

  1;

  2,  1;

  3,  4,  2;

  6, 10,  8,  3;

  9, 24, 27, 16,  5;

First three polynomials u(n,x):

  1

  2 + x

  3 + 4x + 2x^2.

From Philippe Deléham, Mar 29 2012: (Start)

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

   1;

   1,  0;

   2,  1,  0;

   3,  4,  2,  0;

   6, 10,  8,  3,  0;

   9, 24, 27, 16,  5,  0;

  18, 51, 74, 62, 30,  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 = 1; 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[%]   (* A210793 *)

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

TableForm[cv]

Flatten[%]   (* A210794 *)

Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A000244 *)

Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A000244 *)

Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A000012 *)

Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A077925 *)

CROSSREFS

Cf. A210794, A208510.

Sequence in context: A209137 A269752 A122164 * A281715 A076632 A105646

Adjacent sequences:  A210790 A210791 A210792 * A210794 A210795 A210796

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 26 2012

STATUS

approved

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Last modified January 25 08:01 EST 2021. Contains 340416 sequences. (Running on oeis4.)