A175386 -id:A175386 - cp's OEIS frontend

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).

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

A175385 a(n) = numerator of Sum_{i=1..n} binomial(2n-i-1,i-1)/i.

Original entry on oeis.org

1, 3, 17, 23, 61, 107, 421, 1103, 5777, 7563, 19801, 103681, 135721, 355323, 1860497, 2435423, 6376021, 11128427, 43701901, 114413063, 599074577, 784198803, 2053059121, 10749957121, 14071876561, 36840651123, 192900153617

Offset: 1

Vladimir Shevelev, Zak Seidov, Apr 24 2010

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

Cf. A175386 (denominator).

Table[Numerator[Sum[(1/i)*Binomial[2n-i-1,i-1],{i,1,n}]],{n,1,50}]

Sum_{i=1..n} C(2n-i-1,i-1)/i = (2F1(1/2-n,-n;1-2 n;-4) -1)/(2n), where 2F1 is the Gaussian Hypergeometric Function.

A242926 a(n) = denominator of B(0,n), where B(n,n) = 0, B(n-1,n) = 1/n and otherwise B(m,n) = B(m-1,n+1) - B(m-1,n).

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 4, 3, 5, 1, 4, 1, 7, 15, 8, 1, 18, 1, 10, 21, 11, 1, 24, 5, 13, 9, 14, 1, 30, 1, 16, 11, 17, 35, 12, 1, 19, 39, 20, 1, 42, 1, 22, 9, 23, 1, 48, 7, 25, 17, 26, 1, 54, 55, 28, 19, 29, 1, 20, 1

Offset: 0

Paul Curtz, May 26 2014

nonn

The numerators are A189731(n).
B(0,n) = 0, 1, 1, 3/2, 2, 17/6, 4, 23/4, 25/3, 61/5, 18, 107/4, 40, 421/7, ...
is a super autosequence as defined in A242563.
The positive integers in B(0,n) give A064723(n). Corresponding rank: A006093(n+1). B(0,n) is linked to the primes A000040.
Divisor of B(0,n), n > 0: 1, 1, 1, 2, 2, 4, 5, ... = A172128(n+1).
Common (LCM) denominators for the antidiagonals: 1, 1, 1, 2, 2, 6, 6, 12, 12, ... = A139550(n+1)?.
1 = 1
1/2 +  3/2 = 2
1/3 +  5/6 + 17/6 = 4
1/4 + 7/12 +  7/4 + 23/4 = 25/3
etc.
The positive terms of the first bisection are the sum of the corresponding antidiagonal terms upon the 0's.
0 followed by A001610(n), i.e., 0, 0, 2, 3, 6, 10, 17, ... is an autosequence of the second kind.

Cf. A000204, A175386, A001610, A189731, A139550, A242563.

Table[Denominator[(LucasL[n+1]-1)/(n+1)], {n, 0, 100}] (* Artur Jasinski, Nov 06 2022 *)

a(2n+1) = A175386(n).
a(n) = denominator(A001610(n)/(n+1)). [edited by Michel Marcus, Nov 14 2022]
a(n) = denominator((A000204(n+1) - 1)/(n+1)). - Artur Jasinski, Nov 06 2022

a(24)-a(60) from Jean-François Alcover, May 26 2014

cp's OEIS Frontend

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).

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A175385 a(n) = numerator of Sum_{i=1..n} binomial(2n-i-1,i-1)/i.

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A242926 a(n) = denominator of B(0,n), where B(n,n) = 0, B(n-1,n) = 1/n and otherwise B(m,n) = B(m-1,n+1) - B(m-1,n).

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