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A094437 Triangular array T(n,k)=F(k+2)C(n,k), k=0,1,2,3,...,n; n>=0. 6
1, 1, 2, 1, 4, 3, 1, 6, 9, 5, 1, 8, 18, 20, 8, 1, 10, 30, 50, 40, 13, 1, 12, 45, 100, 120, 78, 21, 1, 14, 63, 175, 280, 273, 147, 34, 1, 16, 84, 280, 560, 728, 588, 272, 55, 1, 18, 108, 420, 1008, 1638, 1764, 1224, 495, 89, 1, 20, 135, 600, 1680, 3276, 4410, 4080, 2475, 890 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Let F(n) denote the n-th Fibonacci number (A000045). Then n-th row sum of T is F(2n+2) and n-th alternating row sum is -F(n-2).

A094437 is jointly generated with A094436 as a triangular array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+x*v(n-1)x and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x).  See the Mathematica section. [From Clark Kimberling, Feb 26 2012]

Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 28 2012

LINKS

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

FORMULA

From Philippe Deléham, Apr 28 2012: (Start)

As DELTA-triangle T(n,k):

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

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

EXAMPLE

First four rows:

1

1 2

1 4 3

1 6 9 5

sum = 1+6+9+5=21=F(8); alt.sum = 1-6+9-5=-1=-F(1).

T(3,2)=F(4)*C(3,2)=3*3=9.

From Philippe Deléham, Apr 28 2012: (Start)

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

1

1, 0

1, 2, 0

1, 4, 3, 0

1, 6, 9, 5, 0

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

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 13;

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];

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[%]  (* A094436 *)

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

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

TableForm[cv]

Flatten[%]  (* A094437 *)

CROSSREFS

Cf. A094442, A000045, A094436, A094437.

Sequence in context: A152060 A093190 A132191 * A172431 A053123 A107661

Adjacent sequences:  A094434 A094435 A094436 * A094438 A094439 A094440

KEYWORD

nonn,easy,tabl

AUTHOR

Clark Kimberling, May 03 2004

STATUS

approved

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Last modified July 23 20:32 EDT 2019. Contains 325264 sequences. (Running on oeis4.)