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A208518 Triangle of coefficients of polynomials u(n,x) jointly generated with A208519; see the Formula section. 3
1, 1, 1, 1, 3, 2, 1, 6, 7, 3, 1, 10, 16, 14, 5, 1, 15, 30, 40, 28, 8, 1, 21, 50, 90, 93, 53, 13, 1, 28, 77, 175, 238, 203, 99, 21, 1, 36, 112, 308, 518, 588, 428, 181, 34, 1, 45, 156, 504, 1008, 1428, 1380, 873, 327, 55, 1, 55, 210, 780, 1806, 3066, 3690, 3105 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

coefficient of x^(n-1): = Fibonacci(n) = A000045(n)

col 1:  A000012

col 2:  A000217 (triangular numbers)

col 3:  A005581

col 4:  A117662

alternating row sums: signed version of (-1+Fibonacci(n))

LINKS

Table of n, a(n) for n=1..63.

FORMULA

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

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

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

EXAMPLE

First five rows:

1

1...1

1...3....2

1...6....7....3

1...10...16...14...5

First five polynomials u(n,x):

1

1 + x

1 + 3x + 2x^2

1 + 6x + 7x^2 + 3x^3

1 + 10x + 16x^2 + 14x^3 + 5x^4

MATHEMATICA

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

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

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

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[%]   (* A208518 *)

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

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

TableForm[cv]

Flatten[%]   (* A208519 *)

CROSSREFS

Cf. A208519.

Sequence in context: A079513 A060408 A267121 * A139624 A132276 A257558

Adjacent sequences:  A208515 A208516 A208517 * A208519 A208520 A208521

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 28 2012

STATUS

approved

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Last modified October 20 10:45 EDT 2019. Contains 328257 sequences. (Running on oeis4.)