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A132046
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Triangle read by rows: T(n,0) = T(n,n) = 1, and T(n,k) = 2*binomial(n,k) for 1 <= k <= n - 1.
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10
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1, 1, 1, 1, 4, 1, 1, 6, 6, 1, 1, 8, 12, 8, 1, 1, 10, 20, 20, 10, 1, 1, 12, 30, 40, 30, 12, 1, 1, 14, 42, 70, 70, 42, 14, 1, 1, 16, 56, 112, 140, 112, 56, 16, 1, 1, 18, 72, 168, 252, 252, 168, 72, 18, 1, 1, 20, 90, 240, 420, 504, 420, 240, 90, 20, 1
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OFFSET
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0,5
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COMMENTS
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Double the internal elements of Pascal's triangle. - Paul Barry, Jan 07 2009
Coefficients of 2*(x + 1)^n - (x^n + 1) as a triangle (except for the very first term). - Thomas Baruchel, Jun 02 2018
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LINKS
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FORMULA
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T(n,k) = [k<=n] (0^(n + k) + C(n,k)*(2 - 0^(n - k) - 0^k)). - Paul Barry, Sep 19 2008
G.f.: (1 - t - x*t + 3*x*t^2 - x*t^3 - x^2*t^3)/((1 - t)*(1 - x*t)*(1 - t - x*t)).
T(n+3,k+2) = 2*T(n+2,k+2) - T(n+1,k+2) + 2*T(n+2,k+1) - 3*T(n+1,k+1) - T(n+1,k) + T(n,k+1) + T(n,k), except for n = 0 and k = 0. (End)
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EXAMPLE
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First few rows of the triangle are:
1;
1, 1;
1, 4, 1;
1, 6, 6, 1;
1, 8, 12, 8, 1;
1, 10, 20, 20, 10, 1;
1, 12, 30, 40, 30, 12, 1;
1, 14, 42, 70, 70, 42, 14, 1;
...
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MATHEMATICA
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T[n_, k_] := If[n == k || k == 0, 1, If[k <= n, 2 Binomial[n, k], 0]]
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PROG
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(Maxima) T(n, k) := if k = 0 or k = n then 1 else 2*binomial(n, k)$
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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