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A094442 Triangular array T(n,k)=F(n+2-k)C(n,k), 0<=k<=n. 4
1, 2, 1, 3, 4, 1, 5, 9, 6, 1, 8, 20, 18, 8, 1, 13, 40, 50, 30, 10, 1, 21, 78, 120, 100, 45, 12, 1, 34, 147, 273, 280, 175, 63, 14, 1, 55, 272, 588, 728, 560, 280, 84, 16, 1, 89, 495, 1224, 1764, 1638, 1008, 420, 108, 18, 1, 144, 890, 2475, 4080, 4410, 3276, 1680, 600, 135 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Triangle of coefficients of polynomials v(n,x) jointly generated with A094441; see the Formula section.

Column 1:  Fibonacci numbers, A000045

Row sums:  even-indexed Fibonacci numbers

Alternating row sum:  signed Fibonacci numbers

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

Subtriangle of the triangle given by (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 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..64.

FORMULA

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

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

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

T(n,k) = T(n-1, k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1) - T(n-2,k-2), T(1,0) = T(2,1) = 1, T(2,0) = 2 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Apr 02 2012

EXAMPLE

First five rows:

1

2...1

3...4....1

5...9....6....1

8...20...18...8...1

First three polynomials v(n,x): 1, 2 + x, 3 + 4x + x^2.

Contribution from Philippe Deléham, Apr 02 2012: (Start)

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

1

0, 1

0, 2, 1

0, 3, 4, 1

0, 5, 9, 6, 1

0, 8, 20, 18, 8, 1 . (End)

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] + (x + 1)*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[%]    (* A094441 *)

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

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

TableForm[cv]

Flatten[%]    (* A094442 *)

CROSSREFS

Cf. A094437, A094441, A000045.

Sequence in context: A209413 A126198 A055888 * A060642 A306186 A154929

Adjacent sequences:  A094439 A094440 A094441 * A094443 A094444 A094445

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, May 03 2004

STATUS

approved

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Last modified July 20 16:15 EDT 2019. Contains 325185 sequences. (Running on oeis4.)