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A209126 Triangle of coefficients of polynomials u(n,x) jointly generated with A209127; see the Formula section. 3
1, 2, 1, 2, 3, 2, 2, 5, 7, 3, 2, 7, 14, 13, 5, 2, 9, 23, 32, 25, 8, 2, 11, 34, 62, 71, 46, 13, 2, 13, 47, 105, 156, 149, 84, 21, 2, 15, 62, 163, 295, 367, 304, 151, 34, 2, 17, 79, 238, 505, 767, 827, 604, 269, 55, 2, 19, 98, 332, 805, 1435, 1889, 1798, 1177, 475 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

u(n,n)=A000045(n), Fibonacci numbers

alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,...

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

Subtriangle of the triangle given by (1, 1, -2, 1, 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, Mar 21 2012

LINKS

Table of n, a(n) for n=1..65.

FORMULA

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

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

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

Contribution from Philippe Deléham, Mar 21 2012. (Start)

As DELTA-triangle with 0<=k<=n :

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

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

EXAMPLE

First five rows:

1

2...1

2...3...2

2...5...7....3

2...7...14...13...5

First three polynomials u(n,x):

1

2 + x

2 + 3x + 2x^2

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

1

1, 0

2, 1, 0

2, 3, 2, 0

2, 5, 7, 3, 0

2, 7, 14, 13, 5, 0 . Philippe Deléham, Mar 21 2012

MATHEMATICA

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

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

v[n_, x_] := x*u[n - 1, x] + 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[%]    (* A209126 *)

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

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

TableForm[cv]

Flatten[%]    (* A209127 *)

CROSSREFS

Cf. A209127, A208510.

Sequence in context: A155002 A182413 A296664 * A272377 A103342 A275438

Adjacent sequences:  A209123 A209124 A209125 * A209127 A209128 A209129

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 05 2012

STATUS

approved

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Last modified October 20 02:18 EDT 2019. Contains 328244 sequences. (Running on oeis4.)