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A132731
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Triangle T(n,k) = 2 * binomial(n,k) - 2 with T(n,0) = T(n,n) = 1, read by rows.
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4
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1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 10, 6, 1, 1, 8, 18, 18, 8, 1, 1, 10, 28, 38, 28, 10, 1, 1, 12, 40, 68, 68, 40, 12, 1, 1, 14, 54, 110, 138, 110, 54, 14, 1, 1, 16, 70, 166, 250, 250, 166, 70, 16, 1, 1, 18, 88, 238, 418, 502, 418, 238, 88, 18, 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) = 2*binomial(n, k) - 2 with T(n, 0) = T(n, n) = 1.
T(n, k) = 2*A132044(n, k) with T(n, 0) = T(n, n) = 1.
Sum_{k=0..n} T(n, k) = 2^(n+1) - 2*n - [n=0] = A132732(n). (End)
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EXAMPLE
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First few rows of the triangle are:
1;
1, 1;
1, 2, 1;
1, 4, 4, 1;
1, 6, 10, 6, 1;
1, 8, 18, 18, 8, 1;
1, 10, 28, 38, 28, 10, 1;
1, 12, 40, 68, 68, 40, 12, 1;
...
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MATHEMATICA
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T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n, k] - 2];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 14 2021 *)
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PROG
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(PARI) t(n, k) = 2*binomial(n, k) + ((k==0) || (k==n)) - 2*(k<=n); \\ Michel Marcus, Feb 12 2014
(Sage)
def T(n, k): return 1 if (k==0 or k==n) else 2*binomial(n, k) - 2
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 14 2021
(Magma)
T:= func< n, k | k eq 0 or k eq n select 1 else 2*Binomial(n, k) - 2 >;
[T(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel, Feb 14 2021
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CROSSREFS
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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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