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A209144 Triangle of coefficients of polynomials v(n,x) jointly generated with A209143; see the Formula section. 5
1, 3, 6, 1, 12, 5, 24, 16, 1, 48, 44, 7, 96, 112, 30, 1, 192, 272, 104, 9, 384, 640, 320, 48, 1, 768, 1472, 912, 200, 11, 1536, 3328, 2464, 720, 70, 1, 3072, 7424, 6400, 2352, 340, 13, 6144, 16384, 16128, 7168, 1400, 96, 1, 12288, 35840, 39680, 20736 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Alternating row sums: 1,3,5,7,9,11,13,15,17,...

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

Subtriangle of the triangle given by (3,-1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 07 2012

LINKS

Table of n, a(n) for n=1..53.

FORMULA

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

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

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

Contribution from Philippe Deléham, Mar 07 2012 . (Start)

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

T(n,k) = 2*T(n-1,k) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n.

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

Sum_{k, 0<=k<=n} T(n,k)*x^k = A005408(n), A003945(n), A078057(n), A028859(n), A000244(n), A063782(n), A180168(n) for x = -1, 0, 1, 2, 3, 4, 5 respectively . (End).

EXAMPLE

First five rows:

1

3

6....1

12...5

24...16...1

First three polynomials v(n,x): 1, 3, 6 + x.

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

1

3, 0

6, 1, 0

12, 5, 0, 0

24, 16, 1, 0, 0

48, 44, 7, 0, 0, 0

96, 112, 30, 1, 0, 0, 0

192, 272, 104, 9, 0, 0, 0, 0

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_] := u[n - 1, 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[%]    (* A209143 *)

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

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

TableForm[cv]

Flatten[%]    (* A209144 *)

CROSSREFS

Cf. A209143, A208510.

Sequence in context: A210039 A026250 A210033 * A130724 A120229 A266151

Adjacent sequences:  A209141 A209142 A209143 * A209145 A209146 A209147

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Mar 06 2012

STATUS

approved

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Last modified October 18 03:25 EDT 2019. Contains 328135 sequences. (Running on oeis4.)