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A208759 Triangle of coefficients of polynomials u(n,x) jointly generated with A208760; see the Formula section. 3
1, 1, 2, 1, 4, 6, 1, 6, 16, 16, 1, 8, 30, 56, 44, 1, 10, 48, 128, 188, 120, 1, 12, 70, 240, 504, 608, 328, 1, 14, 96, 400, 1080, 1872, 1920, 896, 1, 16, 126, 616, 2020, 4512, 6672, 5952, 2448, 1, 18, 160, 896, 3444, 9352, 17856, 23040, 18192, 6688, 1, 20, 198, 1248, 5488, 17472, 40600, 67776, 77616, 54976, 18272 (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 given by (1, 0, 0, 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

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

FORMULA

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

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

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

From Philippe Deléham, Mar 18 2012. (Start)

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

G.f.: (1-2y*x-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(2,0) = 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...4...6

1...6...16...16

1...8...30...56...44

First five polynomials u(n,x):

1

1 + 2x

1 + 4x + 6x^2

1 + 6x + 16x^2 + 16x^3

1 + 8x + 30x^2 + 56x^3 + 44x^4

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

1

1, 0

1, 2, 0

1, 4, 6, 0

1, 6, 16, 16, 0

1, 8, 30, 56, 44, 0

1, 10, 48, 128, 188, 120, 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 + 1)*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[%]  (* A208759 *)

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

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

TableForm[cv]

Flatten[%]  (* A208760 *)

Rest[CoefficientList[CoefficientList[Series[(1-2*y*x-2*y^2*x^2)/(1-x-2*y*x- 2*y^2*x^2), {x, 0, 20}, {y, 0, 20}], x], y]//Flatten] (* G. C. Greubel, Mar 28 2018 *)

CROSSREFS

Cf. A208760, A208510.

Sequence in context: A208915 A199704 A062344 * A033877 A059369 A199530

Adjacent sequences:  A208756 A208757 A208758 * A208760 A208761 A208762

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 02 2012

EXTENSIONS

Terms a(58) onward added by G. C. Greubel, Mar 28 2018

STATUS

approved

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Last modified October 16 09:29 EDT 2019. Contains 328056 sequences. (Running on oeis4.)