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A027948 Triangular array T read by rows: T(n,k) = t(n,2k+1) for 0 <= k <= n, T(n,n)=1, t given by A027926, n >= 0.

%I A027948 #23 Sep 08 2022 08:44:49
%S A027948 1,1,1,1,2,1,1,3,3,1,1,3,7,4,1,1,3,8,14,5,1,1,3,8,20,25,6,1,1,3,8,21,
%T A027948 46,41,7,1,1,3,8,21,54,97,63,8,1,1,3,8,21,55,133,189,92,9,1,1,3,8,21,
%U A027948 55,143,309,344,129,10,1,1,3,8,21,55,144,364,674,591,175,11,1
%N A027948 Triangular array T read by rows: T(n,k) = t(n,2k+1) for 0 <= k <= n, T(n,n)=1, t given by A027926, n >= 0.
%H A027948 G. C. Greubel, <a href="/A027948/b027948.txt">Rows n = 0..100 of triangle, flattened</a>
%F A027948 T(n,k) = Sum_{j=0..n-k} binomial(n-j, 2*(n-k-j) -1) with T(n,n)=1 in the region n >= 0, 0 <= k <= n. - _G. C. Greubel_, Sep 29 2019
%e A027948 Triangle begins with:
%e A027948   1;
%e A027948   1, 1;
%e A027948   1, 2, 1;
%e A027948   1, 3, 3,  1;
%e A027948   1, 3, 7,  4,  1;
%e A027948   1, 3, 8, 14,  5,  1;
%e A027948   1, 3, 8, 20, 25,  6, 1;
%e A027948   1, 3, 8, 21, 46, 41, 7, 1; ...
%p A027948 T:= proc(n, k)
%p A027948       if k=n then 1
%p A027948       else add(binomial(n-j, 2*(n-k-j)-1), j=0..n-k)
%p A027948       fi
%p A027948     end:
%p A027948 seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Sep 29 2019
%t A027948 T[n_, k_]:= If[k==n, 1, Sum[Binomial[n-j, 2*(n-k-j)-1], {j, 0, n-k}]];
%t A027948 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 29 2019 *)
%o A027948 (PARI) T(n,k) = if(k==n, 1, sum(j=0,n-k, binomial(n-j, 2*(n-k-j)-1)) );
%o A027948 for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Sep 29 2019
%o A027948 (Magma) T:= func< n,k | k eq n select 1 else &+[Binomial(n-j, 2*(n-k-j) -1): j in [0..n-k]] >;
%o A027948 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 29 2019
%o A027948 (Sage)
%o A027948 def T(n, k):
%o A027948     if (k==n): return 1
%o A027948     else: return sum(binomial(n-j, 2*(n-k-j)-1) for j in (0..n-k))
%o A027948 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Sep 29 2019
%o A027948 (GAP)
%o A027948 T:= function(n,k)
%o A027948     if k=n then return 1;
%o A027948     else return Sum([0..n-k], j-> Binomial(n-j, 2*(n-k-j)-1) );
%o A027948     fi;
%o A027948   end;
%o A027948 Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Sep 29 2019
%Y A027948 The row sums of this (slightly extended) bisection of the "Fibonacci array" A027926 are powers of 2, see A027935 for the other bisection.
%K A027948 nonn,tabl
%O A027948 0,5
%A A027948 _Clark Kimberling_
%E A027948 Name edited by _G. C. Greubel_, Sep 29 2019