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A094435 Triangular array read by rows: T(n,k) = Fibonacci(k)*C(n,k), k = 1...n; n>=1.

%I A094435 #11 Sep 08 2022 08:45:13
%S A094435 1,2,1,3,3,2,4,6,8,3,5,10,20,15,5,6,15,40,45,30,8,7,21,70,105,105,56,
%T A094435 13,8,28,112,210,280,224,104,21,9,36,168,378,630,672,468,189,34,10,45,
%U A094435 240,630,1260,1680,1560,945,340,55,11,55,330,990,2310,3696,4290,3465,1870,605,89
%N A094435 Triangular array read by rows: T(n,k) = Fibonacci(k)*C(n,k), k = 1...n; n>=1.
%C A094435 Let F(n) denote the n-th Fibonacci number (A000045). Then n-th row sum of T is F(2n) and n-th alternating row sum is F(n).
%H A094435 G. C. Greubel, <a href="/A094435/b094435.txt">Rows n = 1..100 of triangle, flattened</a>
%F A094435 From _G. C. Greubel_, Oct 30 2019: (Start)
%F A094435 T(n, k) = binomial(n, k)*Fibonacci(k).
%F A094435 Sum_{k=1..n} binomial(n,k)*Fibonacci(k) = Fibonacci(2*n).
%F A094435 Sum_{k=1..n} (-1)^(k-1)*binomial(n,k)*Fibonacci(k) = Fibonacci(n). (End)
%e A094435 First few rows:
%e A094435   1;
%e A094435   2   1;
%e A094435   3   3   2;
%e A094435   4   6   8   3;
%e A094435   5, 10, 20, 15,  5;
%e A094435   6, 15, 40, 45, 30, 8;
%p A094435 with(combinat); seq(seq(binomial(n,k)*fibonacci(k), k=1..n), n=1..12); # _G. C. Greubel_, Oct 30 2019
%t A094435 Table[Fibonacci[k]*Binomial[n, k], {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Oct 30 2019 *)
%o A094435 (PARI) T(n,k) = binomial(n,k)*fibonacci(k);
%o A094435 for(n=1,12, for(k=1,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Oct 30 2019
%o A094435 (Magma) [Binomial(n,k)*Fibonacci(k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Oct 30 2019
%o A094435 (Sage) [[binomial(n,k)*fibonacci(k) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Oct 30 2019
%o A094435 (GAP) Flat(List([1..12], n-> List([1..n], k-> Binomial(n,k)*Fibonacci(k) ))); # _G. C. Greubel_, Oct 30 2019
%Y A094435 Cf. A000045.
%Y A094435 Cf. A094436, A094437, A094438, A094439, A094440, A094441, A094442, A094443, A094444.
%K A094435 nonn,tabl
%O A094435 1,2
%A A094435 _Clark Kimberling_, May 03 2004