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A224270
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Absolute values of the numerators of the third column of ( 0 followed by (mix 0 , A001803(n))/A060818(n) ) and its successive differences.
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2
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1, 1, 5, 11, 95, 203, 861, 1815, 30459, 63635, 264979, 550069, 4555915, 9412543, 38816525, 79898895, 2627302995, 5392044675, 22104436695, 45256266825, 370241638305, 756514878405, 3088866211275, 6300861570705, 102746354288175, 209286947903319
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OFFSET
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0,3
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COMMENTS
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The array is
0, 0, 1, 0, 3/2, 0, 15/8, 0,...
0, 1, -1, 3/2, -3/2, 15/8, -15/8,...
1, -2, 5/2, -3, 27/8, -15/4,...
-3, 9/2, -11/2, 51/8, -57/8,...
15/2, -10, 95/8, -27/2,...
-35/2, 175/8, -203/8,...
315/8, -189/4,...
-693/8,...
Note A001803 in the first column and a variant of A206771(n) in the second column.
Now consider a(n)/A046161(n) and its differences:
1, 1/2, 5/8, 11/16, 95/128, 203/256, 861/1024,...
-1/2, 1/8, 1/16, 7/128, 13/256, 49/1024,... =b(n)/A046161(n)
5/8, -1/16, -1/128, -1/256, -3/1024,...
-11/16, 7/128, 1/256, 1/1024,...
95/128, -13/256, -3/1024,...
-203/256, 49/1024,...
861/1024,...
This an autosequence of second kind. The first column is the signed sequence.
(Its companion, the corresponding autosequence of first kind, is 0, 1, 1, 9/8, 5/4,... in A206771).
b(n) = a(n+1) - A171977*a(n). Also for two successive rows (with shifted A171977).
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LINKS
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FORMULA
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EXAMPLE
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a(n)=numerators of 0+1=1, 0+1/2=1/2, 1/4+3/8=5/8, 3/8+5/16=11/16, 15/32+35/128=95/128,... .
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MATHEMATICA
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nmax = 25; t1 = Table[ Numerator[ (2*n+1)*(Binomial[2*n, n]/4^n)] / Denominator[ Binomial[2*n, n]/4^n], {n, 0, Ceiling[nmax/2]}]; t2 = Join[{0}, Table[ If[ OddQ[n], 0, t1[[n/2]] ], {n, 1, nmax+2}] ]; t3 = Table[ Differences[t2, n], {n, 0, nmax}]; t3[[All, 3]] // Numerator // Abs (* Jean-François Alcover, Apr 02 2013 *)
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CROSSREFS
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KEYWORD
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nonn,frac,less
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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