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A208910 Triangle of coefficients of polynomials v(n,x) jointly generated with A208755; see the Formula section. 2
1, 1, 3, 1, 3, 8, 1, 3, 10, 22, 1, 3, 12, 32, 60, 1, 3, 14, 42, 100, 164, 1, 3, 16, 52, 144, 308, 448, 1, 3, 18, 62, 192, 480, 936, 1224, 1, 3, 20, 72, 244, 680, 1568, 2816, 3344, 1, 3, 22, 82, 300, 908, 2352, 5040, 8400, 9136, 1, 3, 24, 92, 360, 1164, 3296 (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, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -1/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham Apr 01 2012

LINKS

Table of n, a(n) for n=1..62.

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)+1,

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

Contribution from Philippe Deléham, Apr 01 2012. (Start)

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

G.f.: (1-2*y*x+3*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(2,1) = 3, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k<0 or if k>n. (End)

T(n,k) = 2*A208757(n,k) - A208332(n,k). - Philippe Deléham, Apr 15 2012

EXAMPLE

First five rows:

1

1...3

1...3...8

1...3...10...22

1...3...12...32...60]

First five polynomials v(n,x):

1

1 + 3x

1 + 3x + 8x^2

1 + 3x + 10x^2 + 22x^3

1 + 3x + 12x^2 + 32x^3 + 60x^4

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

1

1, 0

1, 3, 0

1, 3, 8, 0

1, 3, 10, 22, 0

1, 3, 12, 32, 60, 0

1, 3, 14, 42, 100, 164, 0 . - Philippe Deléham, Apr 01 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] + 1;

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[%]  (* A208755 *)

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

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

TableForm[cv]

Flatten[%]  (* A208910 *)

CROSSREFS

Cf. A208755, A208510.

Sequence in context: A171843 A132476 A103279 * A209760 A046544 A011088

Adjacent sequences:  A208907 A208908 A208909 * A208911 A208912 A208913

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 03 2012

STATUS

approved

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Last modified October 16 16:13 EDT 2019. Contains 328101 sequences. (Running on oeis4.)