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A210868 Triangle of coefficients of polynomials u(n,x) jointly generated with A210869; see the Formula section. 2
1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 2, 5, 3, 5, 1, 3, 5, 10, 5, 8, 1, 3, 9, 10, 20, 8, 13, 1, 4, 9, 22, 20, 38, 13, 21, 1, 4, 14, 22, 51, 38, 71, 21, 34, 1, 5, 14, 40, 51, 111, 71, 130, 34, 55, 1, 5, 20, 40, 105, 111, 233, 130, 235, 55, 89, 1, 6, 20, 65, 105, 256, 233, 474 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

In row n the first two terms are 1 and floor(n/2), and the last two, for n>1, are F(n-1) and F(n), where F = A000045 (Fibonacci numbers).

Row sums: 1,2,4,8,16,32,...; A000079.

Alternating row sums:  A151575

For a discussion and guide to related arrays, see A208510.

Subtriangle of the triangle given by (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 02 2012

LINKS

Table of n, a(n) for n=1..74.

FORMULA

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

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

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

From Philippe Deléham, Apr 02 2012: (Start)

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

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

T(n,k) = T(n-1,k-1) + T(n-2,k) + 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 six rows:

  1

  1...1

  1...1...2

  1...2...2...3

  1...2...5...3....5

  1...3...5...10...5...8

First three polynomials u(n,x): 1, 1 + x, 1 + x + 2x^2.

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

1

1, 0

1, 1, 0

1, 1, 2, 0

1, 2, 2, 3, 0

1, 2, 5, 3, 5, 0

1, 3, 5, 10, 5, 8, 0. - Philippe Deléham, Apr 02 2012

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 14;

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

v[n_, x_] := (x + n)*u[n - 1, x] + x*v[n - 1, x] - 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[%]   (* A210866 *)

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

TableForm[cv]

Flatten[%]   (* A210867 *)

CROSSREFS

Cf. A210867, A208510.

Sequence in context: A290399 A109400 A202389 * A176853 A261787 A302480

Adjacent sequences:  A210865 A210866 A210867 * A210869 A210870 A210871

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 29 2012

STATUS

approved

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Last modified October 14 06:51 EDT 2019. Contains 327995 sequences. (Running on oeis4.)