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A208345 Triangle of coefficients of polynomials v(n,x) jointly generated with A208344; see the Formula section. 5
1, 0, 3, 0, 1, 7, 0, 1, 3, 17, 0, 1, 3, 10, 41, 0, 1, 3, 11, 30, 99, 0, 1, 3, 12, 35, 87, 239, 0, 1, 3, 13, 40, 108, 245, 577, 0, 1, 3, 14, 45, 130, 322, 676, 1393, 0, 1, 3, 15, 50, 153, 406, 938, 1836, 3363, 0, 1, 3, 16, 55, 177, 497, 1236, 2682, 4925, 8119, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

row sums, u(n,1):  (1,2,5,13,...), odd-indexed Fibonacci numbers

row sums, v(n,1):  (1,3,8,21,...), even-indexed Fibonacci numbers

As triangle T(n,k) with 0<=k<=n, it is (0, 1/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

LINKS

Table of n, a(n) for n=1..68.

FORMULA

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

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

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

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

Sum_{k, 0<=k<=n} T(n,k)*x^k = A152167(n), A000007(n), A001906(n+1), A003948(n) for x = -1, 0, 1, 2 respectively.

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A078057(n), A001906(n+1), A000244(n), A081567(n), A083878(n), A165310(n) for x = 0, 1, 2, 3, 4, 5 respectively. (END) - Philippe Deléham, Feb 26 2012

EXAMPLE

First five rows:

1

0...3

0...1...7

0...1...3...17

0...1...3...10...41

First five polynomials u(n,x):

1, 3x, x + 7x^2, x + 3x^2 + 17x^3, x + 3x^2 + 10x^3 +

41x^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_] := 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[%]    (* A208344 *)

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

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

TableForm[cv]

Flatten[%]    (* A208345 *)

Table[u[n, x] /. x -> 1, {n, 1, z}]

Table[v[n, x] /. x -> 1, {n, 1, z}]

CROSSREFS

Cf. A208344.

Cf. A084938, A000045, A000244, A001906

Sequence in context: A143397 A244118 A273155 * A216807 A216802 A297786

Adjacent sequences:  A208342 A208343 A208344 * A208346 A208347 A208348

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 25 2012

STATUS

approved

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Last modified January 18 01:36 EST 2019. Contains 319260 sequences. (Running on oeis4.)