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A094438 Triangular array T(n,k) = Fibonacci(k+3)*C(n,k), k=0..n, n>=0.

%I A094438 #14 Sep 08 2022 08:45:13
%S A094438 2,2,3,2,6,5,2,9,15,8,2,12,30,32,13,2,15,50,80,65,21,2,18,75,160,195,
%T A094438 126,34,2,21,105,280,455,441,238,55,2,24,140,448,910,1176,952,440,89,
%U A094438 2,27,180,672,1638,2646,2856,1980,801,144,2,30,225,960,2730,5292,7140,6600,4005,1440,233
%N A094438 Triangular array T(n,k) = Fibonacci(k+3)*C(n,k), k=0..n, n>=0.
%C A094438 Let F(n) denote the n-th Fibonacci number (A000045). Then n-th row sum of T is F(2n+3) and n-th alternating row sum is F(n-3).
%H A094438 G. C. Greubel, <a href="/A094438/b094438.txt">Rows n = 0..100 of triangle, flattened</a>
%F A094438 From _G. C. Greubel_, Oct 30 2019: (Start)
%F A094438 T(n, k) = binomial(n,k)*Fibonacci(k+3).
%F A094438 Sum_{k=0..n} T(n,k) = Fibonacci(2*n+3).
%F A094438 Sum_{k=0..n} (-1)^k * T(n,k) = Fibonacci(n-3). (End)
%e A094438 First few rows:
%e A094438   2;
%e A094438   2   3;
%e A094438   2   6   5;
%e A094438   2   9  15   8;
%e A094438   2, 12, 30, 32, 13;
%e A094438   2, 15, 50, 80, 65, 21;
%p A094438 with(combinat); seq(seq(fibonacci(k+3)*binomial(n,k), k=0..n), n=0..12); # _G. C. Greubel_, Oct 30 2019
%t A094438 Table[Fibonacci[k+3]Binomial[n,k],{n,0,12},{k,0,n}]//Flatten (* _Harvey P. Dale_, Dec 16 2017 *)
%o A094438 (PARI) T(n,k) = binomial(n,k)*fibonacci(k+3);
%o A094438 for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Oct 30 2019
%o A094438 (Magma) [Binomial(n,k)*Fibonacci(k+3): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 30 2019
%o A094438 (Sage) [[binomial(n,k)*fibonacci(k+3) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Oct 30 2019
%o A094438 (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(k+3) ))); # _G. C. Greubel_, Oct 30 2019
%Y A094438 Cf. A000045, A094435, A094436, A094437, A094439.
%Y A094438 Cf. A094440, A094441, A094442, A094443, A094444.
%K A094438 nonn,easy,tabl
%O A094438 0,1
%A A094438 _Clark Kimberling_, May 03 2004