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A208332 Triangle of coefficients of polynomials u(n,x) jointly generated with A208333; see the Formula section. 4
1, 1, 1, 1, 1, 4, 1, 1, 6, 10, 1, 1, 8, 16, 28, 1, 1, 10, 22, 52, 76, 1, 1, 12, 28, 80, 156, 208, 1, 1, 14, 34, 112, 256, 472, 568, 1, 1, 16, 40, 148, 376, 832, 1408, 1552, 1, 1, 18, 46, 188, 516, 1296, 2640, 4176, 4240, 1, 1, 20, 52, 232, 676, 1872, 4320 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Subtriangle of the triangle given by (1, 0, -1, 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 09 2012

LINKS

Table of n, a(n) for n=1..63.

FORMULA

u(n,x) = u(n-1,x) + x*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 09 2012: (Start)

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

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

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 2*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, 1, 4;

  1, 1, 6, 10;

  1, 1, 8, 16, 28;

First five polynomials u(n,x):

1, 1 + x, 1 + x + 4x^2, 1 + x + 6x^2 + 10x^3, 1 + x + 8x^2 + 16x^3 + 28x^4.

From Philippe Deléham, Apr 09 2012: (Start)

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

  1;

  1, 0;

  1, 1, 0;

  1, 1, 4, 0;

  1, 1, 6, 10, 0;

  1, 1, 8, 16, 28, 0;

  1, 1, 10, 22, 52, 76, 0;

  1, 1, 12, 28, 80, 156, 208, 0;

  ... (End)

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 13;

u[n_, x_] := u[n - 1, x] + x*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[%]  (* A208332 *)

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

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

TableForm[cv]

Flatten[%]  (* A208333 *)

CROSSREFS

Cf. A208332.

Sequence in context: A174376 A131399 A069322 * A075112 A202687 A046554

Adjacent sequences:  A208329 A208330 A208331 * A208333 A208334 A208335

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 26 2012

STATUS

approved

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Last modified September 18 17:00 EDT 2020. Contains 337170 sequences. (Running on oeis4.)