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A209125 Triangle of coefficients of polynomials u(n,x) jointly generated with A164975; see the Formula section. 3
1, 2, 1, 3, 4, 2, 5, 9, 9, 4, 8, 20, 25, 20, 8, 13, 40, 65, 65, 44, 16, 21, 78, 150, 190, 162, 96, 32, 34, 147, 331, 490, 521, 392, 208, 64, 55, 272, 697, 1192, 1473, 1368, 928, 448, 128, 89, 495, 1425, 2745, 3888, 4185, 3480, 2160, 960, 256, 144, 890 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Alternating row sums:  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, 1, 0, 0, 0, 0, 0, 0, 0, ....) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 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) = u(n-1,x) + 2x*v(n-1,x),

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

From Philippe Deléham, Mar 21 2012: (Start)

As DELTA-triangle with 0 <= k <= n:

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

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

EXAMPLE

First five rows:

  1;

  2,  1;

  3,  4,  2;

  5,  9,  9,  4;

  8, 20, 25, 20,  8;

First three polynomials u(n,x):

  1

  2 + x

  3 + 4x + 2x^2

From Philippe Deléham, Mar 21 2012: (Start)

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

  1;

  1,  0;

  2,  1,  0;

  3,  4,  2,  0;

  5,  9,  9,  4, 0;

  8, 20, 25, 20, 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_] := 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[%]    (* A209125 *)

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

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

TableForm[cv]

Flatten[%]    (* A164975 *)

CROSSREFS

Cf. A164975, A208510.

Sequence in context: A026249 A130527 A026366 * A209137 A269752 A122164

Adjacent sequences:  A209122 A209123 A209124 * A209126 A209127 A209128

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 05 2012

STATUS

approved

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Last modified May 8 09:21 EDT 2021. Contains 343666 sequences. (Running on oeis4.)