|
%I #23 Feb 08 2023 07:46:54
%S 1,1,5,11,95,203,861,1815,30459,63635,264979,550069,4555915,9412543,
%T 38816525,79898895,2627302995,5392044675,22104436695,45256266825,
%U 370241638305,756514878405,3088866211275,6300861570705,102746354288175,209286947903319
%N Absolute values of the numerators of the third column of ( 0 followed by (mix 0 , A001803(n))/A060818(n) ) and its successive differences.
%C The array is
%C 0, 0, 1, 0, 3/2, 0, 15/8, 0,...
%C 0, 1, -1, 3/2, -3/2, 15/8, -15/8,...
%C 1, -2, 5/2, -3, 27/8, -15/4,...
%C -3, 9/2, -11/2, 51/8, -57/8,...
%C 15/2, -10, 95/8, -27/2,...
%C -35/2, 175/8, -203/8,...
%C 315/8, -189/4,...
%C -693/8,...
%C Note A001803 in the first column and a variant of A206771(n) in the second column.
%C Now consider a(n)/A046161(n) and its differences:
%C 1, 1/2, 5/8, 11/16, 95/128, 203/256, 861/1024,...
%C -1/2, 1/8, 1/16, 7/128, 13/256, 49/1024,... =b(n)/A046161(n)
%C 5/8, -1/16, -1/128, -1/256, -3/1024,...
%C -11/16, 7/128, 1/256, 1/1024,...
%C 95/128, -13/256, -3/1024,...
%C -203/256, 49/1024,...
%C 861/1024,...
%C This an autosequence of second kind. The first column is the signed sequence.
%C (Its companion, the corresponding autosequence of first kind, is 0, 1, 1, 9/8, 5/4,... in A206771).
%C Main diagonal: 1, 1/8, -1/128,... = A002596(n)/A061549(n) ?
%C b(n) = a(n+1) - A171977*a(n). Also for two successive rows (with shifted A171977).
%H Vincenzo Librandi, <a href="/A224270/b224270.txt">Table of n, a(n) for n = 0..200</a>
%F Numerators of (0, 0 followed by A001803(n)/(4*A046161(n))) + A001790(n)/A046161(n).
%e 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,... .
%t 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 *)
%Y Cf. A098597
%K nonn,frac,less
%O 0,3
%A _Paul Curtz_, Apr 02 2013
%E More terms from _Jean-François Alcover_, Apr 02 2013
|
