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A027948
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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.
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13
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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, 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, 55, 143, 309, 344, 129, 10, 1, 1, 3, 8, 21, 55, 144, 364, 674, 591, 175, 11, 1
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OFFSET
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0,5
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LINKS
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FORMULA
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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
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EXAMPLE
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Triangle begins with:
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, 46, 41, 7, 1; ...
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MAPLE
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T:= proc(n, k)
if k=n then 1
else add(binomial(n-j, 2*(n-k-j)-1), j=0..n-k)
fi
end:
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MATHEMATICA
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T[n_, k_]:= If[k==n, 1, Sum[Binomial[n-j, 2*(n-k-j)-1], {j, 0, n-k}]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 29 2019 *)
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PROG
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(PARI) T(n, k) = if(k==n, 1, sum(j=0, n-k, binomial(n-j, 2*(n-k-j)-1)) );
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Sep 29 2019
(Magma) T:= func< n, k | k eq n select 1 else &+[Binomial(n-j, 2*(n-k-j) -1): j in [0..n-k]] >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 29 2019
(Sage)
def T(n, k):
if (k==n): return 1
else: return sum(binomial(n-j, 2*(n-k-j)-1) for j in (0..n-k))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Sep 29 2019
(GAP)
T:= function(n, k)
if k=n then return 1;
else return Sum([0..n-k], j-> Binomial(n-j, 2*(n-k-j)-1) );
fi;
end;
Flat(List([0..12], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Sep 29 2019
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CROSSREFS
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The row sums of this (slightly extended) bisection of the "Fibonacci array" A027926 are powers of 2, see A027935 for the other bisection.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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