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A242735
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Array read by antidiagonals: form difference table of the sequence of rationals 0, 0 followed by A001803(n)/A046161(n), then extract numerators.
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0
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0, 0, 0, 1, 1, 1, -3, -1, 1, 3, 15, 3, -1, 3, 15, -35, -5, 1, -1, 5, 35, 315, 35, -5, 3, -5, 35, 315, -693, -63, 7, -3, 3, -7, 63, 693, 3003, 231, -21, 7, -5, 7, -21, 231, 3003, -6435, -429, 33, -9, 5, -5, 9, -33, 429, 6435
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OFFSET
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0,7
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COMMENTS
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Difference table of c(n)/d(n) = 0, 0, followed by A001803(n)/A046161(n):
0, 0, 1, 3/2, 15/8, 35/16, 315/128, ...
0, 1, 1/2, 3/8, 5/16, 35/128, 63/256, ...
1, -1/2, -1/8, -1/16, -5/128, -7/256, -21/1024, ...
-3/2, 3/8, 1/16, 3/128, 3/256, 7/1024, 9/2048, ...
15/8, -5/16, -5/128, -3/256, -5/1024, -5/2048, -45/32768, ...
-35/16, 35/128, 7/256, 7/1024, 5/2048, 35/32768, 35/65536, ... etc.
d(n) = 1, 1, followed by A046161(n).
c(n)/d(n) is an autosequence (a sequence whose inverse binomial transform is the signed sequence) of the second kind (the main diagonal is equal to the first upper diagonal multiplied by 2). See A187791.
Antidiagonal denominators: repeat n+1 times d(n).
Third row: Lorentz beta factor = 1 followed by -A098597(n). Lorbeta(n) in A206771.
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LINKS
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EXAMPLE
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a(n) as a triangle:
0;
0, 0;
1, 1, 1;
-3, -1, 1, 3;
15, 3, -1, 3, 15;
etc.
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MATHEMATICA
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c[n_] := (2*n-3)*Binomial[2*(n-2), n-2]/4^(n-2) // Numerator; d[n_] := Binomial[2*(n-2), n-2]/4^(n-2) // Denominator; Clear[a]; a[0, k_] := c[k]/d[k]; a[n_, k_] := a[n, k] = a[n-1, k+1] - a[n-1, k]; Table[a[n-k, k] // Numerator, {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 17 2014 *)
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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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