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A208757 Triangle of coefficients of polynomials u(n,x) jointly generated with A208758; see the Formula section. 4
1, 1, 2, 1, 2, 6, 1, 2, 8, 16, 1, 2, 10, 24, 44, 1, 2, 12, 32, 76, 120, 1, 2, 14, 40, 112, 232, 328, 1, 2, 16, 48, 152, 368, 704, 896, 1, 2, 18, 56, 196, 528, 1200, 2112, 2448, 1, 2, 20, 64, 244, 712, 1824, 3840, 6288, 6688, 1, 2, 22, 72, 296, 920, 2584, 6144 (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 (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012

LINKS

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

FORMULA

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

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

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

Contribution from Philippe Deléham, Mar 18 2012 . (Start)

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

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

EXAMPLE

First five rows:

1

1...2

1...2...6

1...2...8....16

1...2...10...24...44

First five polynomials u(n,x):

1

1 + 2x

1 + 2x + 6x^2

1 + 2x + 8x^2 + 16x^3

1 + 2x + 10x^2 + 24x^3 + 44x^4

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

1

1, 0

1, 2, 0

1, 2, 6, 0

1, 2, 8, 16, 0

1, 2, 10, 24, 44, 0

1, 2, 12, 32, 76, 120, 0

1, 2, 14, 40, 112, 232, 328, 0. Philippe Deléham, Mar 18 2012

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*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[%]  (* A208757 *)

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

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

TableForm[cv]

Flatten[%]  (* A208758 *)

CROSSREFS

Cf. A208758, A208510.

Sequence in context: A096179 A166350 A210227 * A133643 A008305 A208763

Adjacent sequences:  A208754 A208755 A208756 * A208758 A208759 A208760

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 02 2012

STATUS

approved

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Last modified October 14 04:44 EDT 2019. Contains 327995 sequences. (Running on oeis4.)