A094437 - OEIS

%I #25 Sep 08 2022 08:45:13

%S 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,

%T 21,1,14,63,175,280,273,147,34,1,16,84,280,560,728,588,272,55,1,18,

%U 108,420,1008,1638,1764,1224,495,89,1,20,135,600,1680,3276,4410,4080,2475,890

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

%C 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).

%C 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. [_Clark Kimberling_, Feb 26 2012]

%C 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

%H G. C. Greubel, <a href="/A094437/b094437.txt">Rows n = 0..100 of triangle, flattened</a>

%F From _Philippe Deléham_, Apr 28 2012: (Start)

%F As DELTA-triangle T(n,k):

%F 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).

%F 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)

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

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

%e First four rows:

%e   1;

%e   1 2;

%e   1 4 3;

%e   1 6 9 5;

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

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

%e From _Philippe Deléham_, Apr 28 2012: (Start)

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

%e   1;

%e   1, 0;

%e   1, 2,  0;

%e   1, 4,  3,  0;

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

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

%p with(combinat); seq(seq(fibonacci(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 = 13;

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

%t v[n_, x_] := 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[%]  (* A094436 *)

%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[%]  (* A094437 *)

%t (* Second program *)

%t Table[Fibonacci[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(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(k+2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 30 2019

%o (Sage) [[binomial(n,k)*fibonacci(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(k+2) ))); # _G. C. Greubel_, Oct 30 2019

%Y Cf. A000045.

%Y Cf. A094435, A094436, A094438, A094439, A094440, A094441, A094442, A094443, A094444.

%K nonn,easy,tabl

%O 0,3

%A _Clark Kimberling_, May 03 2004