A175385 -id:A175385 - cp's OEIS frontend

A175386 a(n) = denominator of sum((1/i)*C(2n-i-1,i-1),i=1..n).

1, 2, 6, 4, 5, 4, 7, 8, 18, 10, 11, 24, 13, 14, 30, 16, 17, 12, 19, 20, 42, 22, 23, 48, 25, 26, 54, 28, 29, 20, 31, 32, 66, 34, 35, 72, 37, 38, 78, 40, 41, 28, 43, 44, 90, 46, 47, 96, 49, 50, 6, 52, 53, 36, 55, 56, 114, 58, 59, 120, 61, 62, 126, 64, 65, 44, 67, 68, 138, 70, 71

Offset: 1

Zak Seidov, Vladimir Shevelev, Apr 24 2010

We conjecture that sum((1/i)*C(2n-i-1,i-1),i=1..n) is not an integer for n>1.

Cf. A175385.

Table[Denominator[Sum[(1/i)*Binomial[2n-i-1,i-1],{i,1,n}]],{n,1,150}]

According to Mathematica, sum((1/i)*C(2n-i-1,i-1), i=1..n)=

(Hypergeometric2F1[1/2-n,-n,1-2 n,-4]-1)/(2 n).

A189731 a(n) = numerator of B(0,n) where B(n,n) = 0, B(n-1,n) = 1/n, and B(m,n) = B(m-1,n+1) - B(m-1,n).

Original entry on oeis.org

0, 1, 1, 3, 2, 17, 4, 23, 25, 61, 18, 107, 40, 421, 1363, 1103, 210, 5777, 492, 7563, 24475, 19801, 2786, 103681, 33552, 135721, 146401, 355323, 39650, 1860497, 97108, 2435423, 2627065, 6376021, 20633238, 11128427, 1459960, 43701901

Offset: 0

Paul Curtz, Apr 26 2011

Square array B(m,n) begins:
    0,    1/1,   1/1,   3/2,   2/1,  17/6,  ...
   1/1,    0,    1/2,   1/2,   5/6,   7/6,  ...
  -1/1,   1/2,    0,    1/3,   1/3,   7/12, ...
   3/2,  -1/2,   1/3,    0,    1/4,   1/4,  ...
  -2/1,   5/6,  -1/3,   1/4,    0,    1/5,  ...
  17/6,  -7/6,   7/12, -1/4,   1/5,    0,   ...
The inverse binomial transform of B(0,n) gives B(n,0) and thus it is an eigensequence in the sense that it remains the same (up to a sign) under inverse binomial transform.
The bisection of B(0,n) (odd part) gives A175385/A175386, and thus a(2*n+1) = A175385(n+1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000204, A242926 (denominators).

Cf. A174341, A177690, A181722.

B:= proc(m, n) option remember;
      if m=n then 0
    elif n=m+1 then 1/n
    elif n>m then B(m, n-1) +B(m+1, n-1)
    else B(m-1, n+1) -B(m-1, n)
      fi
    end:
a:= n-> numer(B(0, n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 29 2011

Rest[Numerator[Abs[CoefficientList[Normal[Series[Log[1 - x^2/(1 + x)], {x, 0, 40}]], x]]]] (* Vaclav Kotesovec, Jul 07 2020 *)
Table[Numerator[(LucasL[n]-1)/n],{n,1,38}] (* Artur Jasinski, Oct 21 2022 *)

Numerator of (A000204(n) - 1)/n. - Artur Jasinski, Oct 21 2022

cp's OEIS Frontend

A175386 a(n) = denominator of sum((1/i)*C(2n-i-1,i-1),i=1..n).

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