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A209133 Triangle of coefficients of polynomials u(n,x) jointly generated with A209134; see the Formula section. 3
1, 2, 1, 2, 5, 4, 2, 9, 18, 10, 2, 13, 40, 56, 28, 2, 17, 70, 154, 176, 76, 2, 21, 108, 320, 564, 540, 208, 2, 25, 154, 570, 1344, 1976, 1640, 568, 2, 29, 208, 920, 2700, 5304, 6720, 4928, 1552, 2, 33, 270, 1386, 4848, 11844, 20016, 22320, 14688, 4240 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

Subtriangle of the triangle given by (1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 3, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 10 2012

LINKS

Table of n, a(n) for n=1..55.

FORMULA

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

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

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

From Philippe Deléham, Apr 10 2012: (Start)

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

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

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

  2,  5,  4;

  2,  9, 18, 10;

  2, 13, 40, 56, 28;

First three polynomials u(n,x):

  1

  2 + x

  2 + 5x + 4x^2

From Philippe Deléham, Apr 10 2012: (Start)

(1, 1, -2, 1, 0, 0, 0, ...) DELTA (0, 1, 3, -2, 0, 0, 0, ...) begins:

  1;

  1,   0;

  2,   1,   0;

  2,   5,   4,   0;

  2,   9,  18,  10,   0;

  2,  13,  40,  56,  28,   0;

  2,  17,  70, 154, 176,  76,   0;

  2,  21, 108, 320, 564, 540, 208,  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_] := 2 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[%]    (* A209133 *)

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

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

TableForm[cv]

Flatten[%]    (* A209134 *)

CROSSREFS

Cf. A209134, A208510.

Sequence in context: A121435 A137156 A136457 * A078016 A078046 A319200

Adjacent sequences:  A209130 A209131 A209132 * A209134 A209135 A209136

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 05 2012

STATUS

approved

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Last modified September 26 09:15 EDT 2021. Contains 347664 sequences. (Running on oeis4.)