A224270 - OEIS

%I #34 Oct 13 2024 11:31:02

%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 (interleave 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. A001790, A001803, A002596, A046161, A061549, A098597, A171977, A206771.

%K nonn,frac,less

%O 0,3

%A _Paul Curtz_, Apr 02 2013

%E More terms from _Jean-François Alcover_, Apr 02 2013