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A208329 Triangle of coefficients of polynomials v(n,x) jointly generated with A208328; see the Formula section. 3
1, 0, 3, 0, 2, 5, 0, 2, 4, 11, 0, 2, 4, 14, 21, 0, 2, 4, 18, 32, 43, 0, 2, 4, 22, 44, 82, 85, 0, 2, 4, 26, 56, 130, 188, 171, 0, 2, 4, 30, 68, 186, 324, 438, 341, 0, 2, 4, 34, 80, 250, 492, 834, 984, 683, 0, 2, 4, 38, 92, 322, 692, 1374, 2028, 2202, 1365, 0, 2, 4, 42 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

row sums, u(n,1):  A000129

row sums, v(n,1):  A001333

As triangle T(n,k) with 0<=k<=n, it is (0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 27 2012

LINKS

Table of n, a(n) for n=1..70.

FORMULA

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

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

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

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

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

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

Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A108411(n+1), A000007(n), A001333(n+1) for x = -1, 0, 1 respectively. (END)- Philippe Deléham, Feb 27 2012

EXAMPLE

First five rows:

1

0...3

0...2...5

0...2...4...11

0...2...4...14...21

First five polynomials u(n,x):

1, 3x, 2x + 5x^2, 2x + 4x^2 + 11x^3, 2x + 4x^2 + 14x^3 + 21x^4.

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 13;

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

v[n_, x_] := 2 x*u[n - 1, x] + 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[%]  (* A208328 *)

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

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

TableForm[cv]

Flatten[%]  (* A208329 *)

CROSSREFS

Cf. A208328.

Sequence in context: A273084 A261163 A292244 * A283025 A089598 A117139

Adjacent sequences:  A208326 A208327 A208328 * A208330 A208331 A208332

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 26 2012

STATUS

approved

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Last modified October 19 21:26 EDT 2019. Contains 328236 sequences. (Running on oeis4.)