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Triangular array T read by rows: T(n,k)=t(n,2k), t given by A027926; 0 <= k <= n, n >= 0.
15

%I #15 Sep 08 2022 08:44:49

%S 1,1,1,1,2,1,1,2,4,1,1,2,5,7,1,1,2,5,12,11,1,1,2,5,13,26,16,1,1,2,5,

%T 13,33,51,22,1,1,2,5,13,34,79,92,29,1,1,2,5,13,34,88,176,155,37,1,1,2,

%U 5,13,34,89,221,365,247,46,1,1,2,5,13,34,89,232,530,709,376,56,1

%N Triangular array T read by rows: T(n,k)=t(n,2k), t given by A027926; 0 <= k <= n, n >= 0.

%H G. C. Greubel, <a href="/A027935/b027935.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n,k) = Sum_{j=0..floor((2*n-2*k-1)/2)} binomial(n-j, 2*(n-k-j)). - _G. C. Greubel_, Sep 27 2019

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 2, 4, 1:

%e 1, 2, 5, 7, 1;

%e 1, 2, 5, 12, 11, 1;

%e 1, 2, 5, 13, 26, 16, 1;

%e ...

%p T:= proc(n, k) option remember;

%p add( binomial(n-j, 2*(n-k-j)), j=0..floor((2*n - 2*k+1)/2))

%p end:

%p seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Sep 27 2019

%t T[n_, k_]:= Sum[Binomial[n-j, 2*(n-k-j)], {j,0,Floor[(2*n-2*k+1)/2]}]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 27 2019 *)

%o (PARI) T(n,k) = sum(j=0,(2*n-2*k+1)\2, binomial(n-j, 2*(n-k-j)));

%o for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Sep 27 2019

%o (Magma) T:= func< n,k | &+[Binomial(n-j, 2*(n-k-j)) : j in [0..Floor((2*n -2*k+1)/2)]] >;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 27 2019

%o (Sage) [[ sum(binomial(n-j, 2*(n-k-j)) for j in (0..floor((2*n-2*k+1)/2)) ) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Sep 27 2019

%o (GAP) Flat(List([0..12], n-> List([0..n], k-> Sum([0..Int((2*n-2*k+1)/2) ], j-> Binomial(n-j, 2*(n-k-j))) ))); # _G. C. Greubel_, Sep 27 2019

%Y The row sums of this bisection of the "Fibonacci array" A027926 are powers of 2, see A027948 for the other bisection.

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_