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A209172 Triangle of coefficients of polynomials u(n,x) jointly generated with A209413; see the Formula section. 3
1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 6, 11, 15, 1, 1, 7, 23, 26, 31, 1, 1, 9, 30, 72, 57, 63, 1, 1, 10, 48, 102, 201, 120, 127, 1, 1, 12, 58, 198, 303, 522, 247, 255, 1, 1, 13, 82, 256, 699, 825, 1291, 502, 511, 1, 1, 15, 95, 420, 955, 2223, 2116, 3084, 1013, 1023, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

For n>1, n-th alternating row sum = ((-1)^n)*F(2n-4), where F=A000045 (Fibonacci numbers).  For a discussion and guide to related arrays, see A208510.

Subtriangle of the triangle given by (1, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 11 2012

LINKS

Table of n, a(n) for n=1..66.

FORMULA

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

v(n,x)=u(n-1,x)+2x*v(n-1,x),

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

Contribution from Philippe Deléham, Mar 11 2012. (Start)

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

T(n,k) = 3*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-2) with 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.

G.f.: (1+x-3*y*x-2*y*x^2+2*y^2*x^2)/(1-3*y*x-x^2+2*y^2*x^2). (End)

EXAMPLE

First five rows:

1

1...1

1...3...1

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

1...6...11...15...1

First three polynomials v(n,x): 1, 1 + x, 1 + 3x + x^2.

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

1

1, 0

1, 1, 0

1, 3, 1, 0

1, 4, 7, 1, 0

1, 6, 11, 15, 1, 0

1, 7, 23, 26, 31, 1, 0

1, 9, 30, 72, 57, 63, 1, 0

MATHEMATICA

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

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

v[n_, 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[%]    (* A209172 *)

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

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

TableForm[cv]

Flatten[%]    (* A209413 *)

CROSSREFS

Cf. A209413, A208510.

Sequence in context: A209415 A058879 A208344 * A263950 A160870 A025255

Adjacent sequences:  A209169 A209170 A209171 * A209173 A209174 A209175

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 08 2012

STATUS

approved

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Last modified October 23 10:07 EDT 2019. Contains 328345 sequences. (Running on oeis4.)