A208345 Triangle of coefficients of polynomials v(n,x) jointly generated with A208344; see the Formula section.

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

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: A339350 A244118 A273155 * A216807 A216802 A350248

Adjacent sequences: A208342 A208343 A208344 * A208346 A208347 A208348

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 25 2012

Status

approved