This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A094437 #25 Sep 08 2022 08:45:13
%S A094437 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 A094437 21,1,14,63,175,280,273,147,34,1,16,84,280,560,728,588,272,55,1,18,
%U A094437 108,420,1008,1638,1764,1224,495,89,1,20,135,600,1680,3276,4410,4080,2475,890
%N A094437 Triangular array T(n,k) = Fibonacci(k+2)*C(n,k), k=0..n, n>=0.
%C A094437 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 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 A094437 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 A094437 G. C. Greubel, <a href="/A094437/b094437.txt">Rows n = 0..100 of triangle, flattened</a>
%F A094437 From _Philippe Deléham_, Apr 28 2012: (Start)
%F A094437 As DELTA-triangle T(n,k):
%F A094437 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 A094437 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 A094437 From _G. C. Greubel_, Oct 30 2019: (Start)
%F A094437 T(n, k) = binomial(n, k)*Fibonacci(k+2).
%F A094437 Sum_{k=0..n} T(n,k) = Fibonacci(2*n+2).
%F A094437 Sum_{k=0..n} (-1)^(k+1) * T(n,k) = Fibonacci(n-2). (End)
%e A094437 First four rows:
%e A094437 1;
%e A094437 1 2;
%e A094437 1 4 3;
%e A094437 1 6 9 5;
%e A094437 sum = 1+6+9+5=21=F(8); alt.sum = 1-6+9-5=-1=-F(1).
%e A094437 T(3,2)=F(4)*C(3,2)=3*3=9.
%e A094437 From _Philippe Deléham_, Apr 28 2012: (Start)
%e A094437 (1, 0, 0, 1, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, ...) begins :
%e A094437 1;
%e A094437 1, 0;
%e A094437 1, 2, 0;
%e A094437 1, 4, 3, 0;
%e A094437 1, 6, 9, 5, 0;
%e A094437 1, 8, 18, 20, 8, 0; . (End)
%p A094437 with(combinat); seq(seq(fibonacci(k+2)*binomial(n,k), k=0..n), n=0..12); # _G. C. Greubel_, Oct 30 2019
%t A094437 (* First program *)
%t A094437 u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t A094437 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A094437 v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A094437 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A094437 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A094437 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A094437 TableForm[cu]
%t A094437 Flatten[%] (* A094436 *)
%t A094437 Table[Expand[v[n, x]], {n, 1, z}]
%t A094437 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A094437 TableForm[cv]
%t A094437 Flatten[%] (* A094437 *)
%t A094437 (* Second program *)
%t A094437 Table[Fibonacci[k+2]*Binomial[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 30 2019 *)
%o A094437 (PARI) T(n,k) = binomial(n,k)*fibonacci(k+2);
%o A094437 for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Oct 30 2019
%o A094437 (Magma) [Binomial(n,k)*Fibonacci(k+2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 30 2019
%o A094437 (Sage) [[binomial(n,k)*fibonacci(k+2) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Oct 30 2019
%o A094437 (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(k+2) ))); # _G. C. Greubel_, Oct 30 2019
%Y A094437 Cf. A000045.
%Y A094437 Cf. A094435, A094436, A094438, A094439, A094440, A094441, A094442, A094443, A094444.
%K A094437 nonn,easy,tabl
%O A094437 0,3
%A A094437 _Clark Kimberling_, May 03 2004