A094443 - OEIS

%I #7 Sep 08 2022 08:45:13

%S 2,3,2,5,6,2,8,15,9,2,13,32,30,12,2,21,65,80,50,15,2,34,126,195,160,

%T 75,18,2,55,238,441,455,280,105,21,2,89,440,952,1176,910,448,140,24,2,

%U 144,801,1980,2856,2646,1638,672,180,27,2,233,1440,4005,6600,7140,5292,2730,960,225,30,2

%N Triangular array T(n,k) = Fibonacci(n+3-k)*C(n,k), k=0..n, n>=0.

%H G. C. Greubel, <a href="/A094443/b094443.txt">Rows n = 0..100 of triangle, flattened</a>

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

%F T(n,k) = binomial(n,k)*Fibonacci(n-k+3).

%F Sum_{k=0..n} T(n,k) = Fibonacci(2*n+3).

%F Sum_{k=0..n} (-1)^k * T(n,k) = (-1)^n * Fibonacci(n-3). (End)

%e First few rows:

%e    2;

%e    3,  2;

%e    5,  6,  2;

%e    8, 15,  9,  2;

%e   13, 32, 30, 12,  2;

%e   21, 65, 80, 50, 15, 2;

%p with(combinat): seq(seq(fibonacci(n-k+3)*binomial(n,k), k=0..n), n=0..12); # _G. C. Greubel_, Oct 30 2019

%t Table[Fibonacci[n-k+3]*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+3);

%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+3): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 30 2019

%o (Sage) [[binomial(n,k)*fibonacci(n-k+3) 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+3) ))); # _G. C. Greubel_, Oct 30 2019

%Y Cf. A000045, A094435, A094436, A094437, A094438, A094439, A094440, A094441, A094442, A094444.

%K nonn,tabl

%O 0,1

%A _Clark Kimberling_, May 03 2004