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A208749 Triangle of coefficients of polynomials u(n,x) jointly generated with A208750; see the Formula section. 3
1, 1, 2, 1, 6, 2, 1, 12, 10, 4, 1, 20, 32, 24, 4, 1, 30, 80, 88, 36, 8, 1, 42, 170, 256, 180, 72, 8, 1, 56, 322, 644, 660, 384, 104, 16, 1, 72, 560, 1456, 1992, 1568, 704, 192, 16, 1, 90, 912, 3024, 5256, 5360, 3392, 1344, 272, 32, 1, 110, 1410, 5856, 12552 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

Subtriangle of the triangle T(n,k) given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 14 2012

LINKS

Table of n, a(n) for n=1..60.

FORMULA

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

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

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

As DELTA-triangle: g.f.: (1-x-2*y^2*x^2)/(1-2*x+x^2-2*y*x^2-2*y^2*x^2). - Philippe Deléham, Mar 14 2012

Recurrence: T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = 1, T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > =n. - Philippe Deléham, Mar 14 2012

EXAMPLE

First five rows:

  1;

  1,  2;

  1,  6,  2;

  1, 12, 10,  4;

  1, 20, 32, 24,  4;

First five polynomials u(n,x):

  1

  1 +  2x

  1 +  6x +  2x^2

  1 + 12x + 10x^2 +  4x^3

  1 + 20x + 32x^2 + 24x^3 + 4x^4

From Philippe Deléham, Mar 14 2012: (Start)

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

  1;

  1,  0;

  1,  2,  0;

  1,  6,  2,  0;

  1, 12, 10,  4,  0;

  1, 20, 32, 24,  4,  0;

  1, 30, 80, 88, 36,  8,  0; (End)

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];

v[n_, x_] := (x + 1)*u[n - 1, 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[%]    (* A208749 *)

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

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

TableForm[cv]

Flatten[%]    (* A208750 *)

CROSSREFS

Cf. A208750, A208510.

Sequence in context: A248779 A286030 A208905 * A208751 A133200 A103881

Adjacent sequences:  A208746 A208747 A208748 * A208750 A208751 A208752

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 02 2012

STATUS

approved

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Last modified December 5 00:42 EST 2020. Contains 338943 sequences. (Running on oeis4.)