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A209128 Triangle of coefficients of polynomials u(n,x) jointly generated with A209129; see the Formula section. 3
1, 2, 1, 2, 4, 3, 2, 6, 12, 7, 2, 8, 22, 32, 17, 2, 10, 34, 70, 86, 41, 2, 12, 48, 124, 216, 228, 99, 2, 14, 64, 196, 428, 644, 600, 239, 2, 16, 82, 288, 744, 1408, 1876, 1568, 577, 2, 18, 102, 402, 1188, 2664, 4476, 5364, 4074, 1393, 2, 20, 124, 540, 1786 (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, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012

Row sums are powers of 3 (A000244) .- Philippe Deléham, Mar 21 2012

LINKS

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

FORMULA

u(n,x)=u(n-1,x)+(x+1)*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 21 2012. (Start)

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

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

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

EXAMPLE

First five rows:

1

2...1

2...4...3

2...6...12...7

2...8...22...32...17

First three polynomials u(n,x):

1

2 + x

2 + 4x + 3x^2

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

1

1, 0

2, 1, 0

2, 4, 3, 0

2, 6, 12, 7, 0

2, 8, 22, 32, 17, 0 . Philippe Deléham, Mar 21 2012

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_] := 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[%]    (* A209128 *)

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

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

TableForm[cv]

Flatten[%]    (* A209129 *)

CROSSREFS

Cf. A209129, A208510.

Sequence in context: A133422 A099312 A117505 * A209131 A165053 A302982

Adjacent sequences:  A209125 A209126 A209127 * A209129 A209130 A209131

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 05 2012

STATUS

approved

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Last modified October 23 13:38 EDT 2019. Contains 328345 sequences. (Running on oeis4.)