A094442 Triangular array T(n,k) = Fibonacci(n+2-k)*C(n,k), 0 <= k <= n.

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, 20, 1

Offset

0,2

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

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

Let u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = u(n-1,x) + (x+1)*v(n-1, x), where u(1,x)=1, v(1,x)=1 then the coefficients of the polynomials of v(n,x) produce this sequence.

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

From G. C. Greubel, Oct 30 2019: (Start)

T(n,k) = binomial(n,k)*Fibonacci(n-k+2).

Sum_{k=0..n} T(n,k) = Fibonacci(2*n+2)

Sum_{k=0..n} (-1)^(k+1) * T(n,k) = (-1)^n * Fibonacci(n-2). (End)

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.
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)

Maple

with(combinat); seq(seq(fibonacci(n-k+2)*binomial(n, k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019

Mathematica

(* First program *)
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 *)
(* Second program *)
Table[Fibonacci[n-k+2]*Binomial[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Oct 30 2019 *)

Prog

(PARI) T(n, k) = binomial(n, k)*fibonacci(n-k+2);
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Oct 30 2019
(Magma) [Binomial(n, k)*Fibonacci(n-k+2): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019
(Sage) [[binomial(n, k)*fibonacci(n-k+2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n, k)*Fibonacci(n-k+2) ))); # G. C. Greubel, Oct 30 2019

Crossrefs

Cf. A000045, A094435, A094436, A094437, A094438, A094439, A094440, A094441, A094443, A094444.

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