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A163945
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Triangle interpolating between (-1)^n (A033999) and the swinging factorial function (A056040) restricted to odd indices (2n+1)$ (A002457).
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0
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1, -1, 6, 1, -12, 30, -1, 18, -90, 140, 1, -24, 180, -560, 630, -1, 30, -300, 1400, -3150, 2772, 1, -36, 450, -2800, 9450, -16632, 12012, -1, 42, -630, 4900, -22050, 58212, -84084, 51480, 1, -48, 840, -7840, 44100, -155232, 336336, -411840, 218790
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OFFSET
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0,3
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COMMENTS
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Triangle read by rows.
For n >= 0, k >= 0 let T(n, k) = (-1)^(n-k) binomial(n,k) (2*k+1)$ where i$ denotes the swinging factorial of i (A056040).
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LINKS
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FORMULA
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Conjectural g.f.: sqrt(1 + t)/(1 + (1 - 4*x)*t)^(3/2) = 1 + (-1 + 6*x)*t + (1 - 12*x + 30*x^2)*t^2 + .... - Peter Bala, Nov 10 2013
T(n, k) = ((-1)^(k mod 2) + n)*((2*k + 1)!/(k!)^2)*binomial(n, n - k). - Detlef Meya, Oct 07 2023
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EXAMPLE
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1;
-1, 6;
1, -12, 30;
-1, 18, -90, 140;
1, -24, 180, -560, 630;
-1, 30, -300, 1400, -3150, 2772;
1, -36, 450, -2800, 9450, -16632, 12012;
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MAPLE
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swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n, k) (-1)^(n-k)*binomial(n, k)*swing(2*k+1) end:
seq(print(seq(a(n, k), k=0..n)), n=0..8);
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MATHEMATICA
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T[n_, k_] := ((-1)^(Mod[k, 2]+n)*((2*k+1)!/(k!)^2)*Binomial[n, n-k]);
Flatten[Table[T[n, k], {n, 0, 8}, {k, 0, n}]] (*End*)
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CROSSREFS
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Row sums are the inverse binomial transform of the beta numbers (A163872).
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KEYWORD
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AUTHOR
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STATUS
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approved
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