A224270 Absolute values of the numerators of the third column of ( 0 followed by (interleave 0 , A001803(n))/A060818(n) ) and its successive differences.

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

Offset: 0

Author

Paul Curtz, Apr 02 2013

Keywords

• nonn
• frac
• less

Comments

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).

Main diagonal: 1, 1/8, -1/128,... = A002596(n)/A061549(n) ?

b(n) = a(n+1) - A171977*a(n). Also for two successive rows (with shifted A171977).

Examples

			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,... .
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A001790, A001803, A002596, A046161, A061549, A098597, A171977, A206771.

Programs

Mathematica

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 *)

Formula

Numerators of (0, 0 followed by A001803(n)/(4*A046161(n))) + A001790(n)/A046161(n).

Extensions

More terms from Jean-François Alcover, Apr 02 2013