A094442 - OEIS

%I #20 Sep 08 2022 08:45:13

%S 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,

%T 1,34,147,273,280,175,63,14,1,55,272,588,728,560,280,84,16,1,89,495,

%U 1224,1764,1638,1008,420,108,18,1,144,890,2475,4080,4410,3276,1680,600,135,20,1

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

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

%C Column 1:  Fibonacci numbers, A000045

%C Row sums:  even-indexed Fibonacci numbers

%C Alternating row sum:  signed Fibonacci numbers

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

%C 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

%H G. C. Greubel, <a href="/A094442/b094442.txt">Rows n = 0..100 of triangle, flattened</a>

%F 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.

%F 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

%F From _G. C. Greubel_, Oct 30 2019: (Start)

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

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

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

%e First five rows:

%e   1;

%e   2,  1;

%e   3,  4,  1;

%e   5,  9,  6, 1;

%e   8, 20, 18, 8, 1;

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

%e From _Philippe Deléham_, Apr 02 2012: (Start)

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

%e   1;

%e   0, 1;

%e   0, 2,  1;

%e   0, 3,  4,  1;

%e   0, 5,  9,  6, 1;

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

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

%t (* First program *)

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

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

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

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%]    (* A094441 *)

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

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

%t TableForm[cv]

%t Flatten[%]    (* A094442 *)

%t (* Second program *)

%t Table[Fibonacci[n-k+2]*Binomial[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 30 2019 *)

%o (PARI) T(n,k) = binomial(n,k)*fibonacci(n-k+2);

%o for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Oct 30 2019

%o (Magma) [Binomial(n,k)*Fibonacci(n-k+2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 30 2019

%o (Sage) [[binomial(n,k)*fibonacci(n-k+2) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Oct 30 2019

%o (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(n-k+2) ))); # _G. C. Greubel_, Oct 30 2019

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

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_, May 03 2004