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A209137 Triangle of coefficients of polynomials u(n,x) jointly generated with A209138; see the Formula section. 4
1, 2, 1, 3, 4, 2, 5, 10, 9, 3, 8, 22, 28, 18, 5, 13, 45, 74, 68, 35, 8, 21, 88, 177, 210, 154, 66, 13, 34, 167, 397, 574, 541, 331, 122, 21, 55, 310, 850, 1446, 1656, 1302, 686, 222, 34, 89, 566, 1758, 3434, 4614, 4404, 2982, 1382, 399, 55, 144, 1020 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Every row begins and ends with a Fibonacci number (A000045).

u(n,1) = n-th row sum = 3^(n-1).

Alternating row sums: 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, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 11 2012

Mirror image of triangle in A209138. - Philippe Deléham, Apr 11 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+1)*u(n-1,x) + x*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-y*x-y*x^2-y^2*x^2)/(1-x-y*x-x^2-y*x^2-y^2*x^2).

T(n,k) = T(n-1,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(2,1) = 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,  4,  2;

  5, 10,  9,  3;

  8, 22, 28, 18, 5;

First three polynomials u(n,x):

  1

  2 +  x

  3 + 4x + 2x^2

From Philippe Deléham, Apr 11 2012: (Start)

(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;

   5, 10,  9,  3,  0;

   8, 22, 28, 18,  5,  0;

  13, 45, 74, 68, 35,  8,  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] + 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[%]    (* A209137 *)

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

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

TableForm[cv]

Flatten[%]    (* A209138 *)

CROSSREFS

Cf. A209138, A208510.

Sequence in context: A130527 A026366 A209125 * A269752 A122164 A210793

Adjacent sequences:  A209134 A209135 A209136 * A209138 A209139 A209140

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 05 2012

STATUS

approved

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Last modified May 11 10:12 EDT 2021. Contains 343788 sequences. (Running on oeis4.)