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A208344 Triangle of coefficients of polynomials u(n,x) jointly generated with A208345; see the Formula section. 3
1, 1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 5, 10, 17, 1, 1, 6, 13, 27, 41, 1, 1, 7, 16, 38, 71, 99, 1, 1, 8, 19, 50, 106, 186, 239, 1, 1, 9, 22, 63, 146, 294, 484, 577, 1, 1, 10, 25, 77, 191, 424, 806, 1253, 1393, 1, 1, 11, 28, 92, 241, 577, 1212, 2191, 3229, 3363, 1, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,6

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

Subtriangle of the triangle given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 09 2012

LINKS

Table of n, a(n) for n=1..69.

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.

Contribution from Philippe Deléham, Apr 09 2012. (Start)

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

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

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

EXAMPLE

First five rows:

1

1...1

1...1...3

1...1...4...7

1...1...5...10...17

First five polynomials u(n,x):

1, 1 + x, 1 + x + 3x^2, 1 + x + 4x^2 + 7x^3, 1 + x + 5x^2 + 10x^3 + 17x^4.

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

1

1, 0

1, 1, 0

1, 1, 3, 0

1, 1, 4, 7, 0

1, 1, 5, 10, 17, 0

1, 1, 6, 13, 27, 41, 0 .- Philippe Deléham, Apr 09 2012

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. A208345.

Sequence in context: A105687 A209415 A058879 * A209172 A263950 A160870

Adjacent sequences:  A208341 A208342 A208343 * A208345 A208346 A208347

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 25 2012

STATUS

approved

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Last modified May 19 04:06 EDT 2019. Contains 323377 sequences. (Running on oeis4.)