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A209696 Triangle of coefficients of polynomials v(n,x) jointly generated with A209695; see the Formula section. 4
1, 1, 3, 1, 6, 7, 1, 9, 23, 17, 1, 12, 48, 76, 41, 1, 15, 82, 204, 233, 99, 1, 18, 125, 428, 765, 682, 239, 1, 21, 177, 775, 1907, 2649, 1935, 577, 1, 24, 238, 1272, 4010, 7656, 8680, 5368, 1393, 1, 27, 308, 1946, 7506, 18358, 28548, 27312, 14641, 3363 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Alternating row sums: 1,-2,2,-2,2,-2,2,-2,...

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, 3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 24 2012

Mirror image of triangle in A054458. - Philippe Deléham, Mar 24 2012

LINKS

Table of n, a(n) for n=1..55.

FORMULA

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

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

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

Contribution from Philippe Deléham, Mar 24 2012. (Start)

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

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

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

EXAMPLE

First five rows:

1

1...3

1...6....7

1...9....23...17

1...12...48...76...41

First three polynomials v(n,x): 1, 1 + 3x , 1 + 6x + 7x^2.

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

1

1, 0

1, 3, 0

1, 6, 7, 0

1, 9, 23, 17, 0

1, 12, 48, 76, 41, 0 . - Philippe Deléham, Mar 24 2012

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];

v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*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[%]    (* A209695 *)

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

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

TableForm[cv]

Flatten[%]    (* A209696 *)

CROSSREFS

Cf. A054458, A209695, A208510.

Sequence in context: A124929 A208766 A259454 * A210749 A199662 A280293

Adjacent sequences:  A209693 A209694 A209695 * A209697 A209698 A209699

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 13 2012

STATUS

approved

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Last modified October 18 02:23 EDT 2019. Contains 328135 sequences. (Running on oeis4.)