A051712 -id:A051712 - cp's OEIS frontend

A164848 a(n) = A026741(n)/A051712(n+1).

1, 1, 3, 2, 1, 3, 1, 4, 3, 1, 1, 6, 1, 1, 3, 4, 1, 3, 1, 2, 3, 1, 1, 12, 1, 1, 3, 2, 1, 3, 1, 4, 3, 1, 1, 6, 1, 1, 3, 4, 1, 3, 1, 2, 3, 1, 1, 12, 1, 1, 3, 2, 1, 3, 1, 4, 3, 1, 1, 6, 1, 1, 3, 4, 1, 3, 1, 2, 3, 1, 1, 12, 1, 1, 3, 2, 1, 3, 1, 4, 3, 1, 1, 6, 1, 1, 3, 4, 1, 3, 1, 2, 3, 1, 1, 12, 1, 1, 3, 2, 1, 3, 1, 4, 3

Offset: 1

Paul Curtz, Aug 28 2009

Twice connected to Bernoulli numbers A164555/A027642 via the Akiyama-Tanigawa algorithm.
Conjecture (checked for the first 3000 entries): periodic with a(n+24)=a(n).
Is this a multiplicative function?
Multiplicative because both A026741 and A051712(n+1) are. - Andrew Howroyd, Jul 26 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A026741, A027642, A051712, A164555.

[Gcd(12, n div Gcd(2, n)): n in [1..100]]; // Vincenzo Librandi, Jul 26 2018

b := proc(n) n/(n+1)/(n+2) ; end: A051712 := proc(n) numer( b(n)-b(n+1)) ; end:
A026741 := proc(n) if type(n,'odd') then n; else n/2; fi; end:
A164848 := proc(n) A026741(n)/A051712(n+1) ; end: seq(A164848(n),n=1..120) ; # R. J. Mathar, Sep 06 2009

Table[GCD[12, n / GCD[2, n]], {n, 100}] (* Vincenzo Librandi, Jul 26 2018 *)

a(n) = gcd(12, n/gcd(2, n)); \\ Andrew Howroyd, Jul 26 2018

a(n) = gcd(12, n/gcd(2, n)). - Andrew Howroyd, Jul 26 2018
From Amiram Eldar, Oct 28 2023: (Start)
Multiplicative with a(2^3) = 2^min(e-1,2), a(3^e) = 3, and a(p^e) = 1 for a prime p >= 5.
Dirichlet g.f.: zeta(s) * (1 + 1/2^(2*s) + 1/2^(3*s-1)) * (1 + 2/3^s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5/2. (End)

Offset set to 1 by R. J. Mathar, Sep 06 2009

A051714 Numerators of table a(n,k) read by antidiagonals: a(0,k) = 1/(k+1), a(n+1,k) = (k+1)*(a(n,k) - a(n,k+1)), n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 3, 1, -1, 1, 1, 2, 1, -1, 0, 1, 1, 5, 2, -3, -1, 1, 1, 1, 3, 5, -1, -1, 1, 0, 1, 1, 7, 5, 0, -4, 1, 1, -1, 1, 1, 4, 7, 1, -1, -1, 1, -1, 0, 1, 1, 9, 28, 49, -29, -5, 8, 1, -5, 5, 1, 1, 5, 3, 8, -7, -9, 5, 7, -5, 5, 0, 1, 1, 11, 15, 27, -28, -343, 295, 200, -44, -1017, 691, -691

Offset: 0

N. J. A. Sloane

Leading column gives the Bernoulli numbers A164555/A027642. - corrected by Paul Curtz, Apr 17 2014

			Table begins:
   1     1/2   1/3    1/4   1/5  1/6  1/7 ...
   1/2   1/3   1/4    1/5   1/6  1/7 ...
   1/6   1/6   3/20   2/15  5/42 ...
   0     1/30  1/20   2/35  5/84 ...
  -1/30 -1/30 -3/140 -1/105 ...
Antidiagonals of numerator(a(n,k)):
  1;
  1,  1;
  1,  1,  1;
  1,  1,  1,  0;
  1,  1,  3,  1, -1;
  1,  1,  2,  1, -1,   0;
  1,  1,  5,  2, -3,  -1,  1;
  1,  1,  3,  5, -1,  -1,  1,  0;
  1,  1,  7,  5,  0,  -4,  1,  1, -1;
  1,  1,  4,  7,  1,  -1, -1,  1, -1,  0;
  1,  1,  9, 28, 49, -29, -5,  8,  1, -5,  5;

Alois P. Heinz, Antidiagonals n = 0..140, flattened
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
Index entries for sequences related to Bernoulli numbers.

Rows 2, 3, 4 give: A026741/A045896, A051712/A051713, A051722/A051723.
Columns 0, 1, 2, 3 give: A000367/A002445, A051716/A051717, A051718/A051719, A051720/A051721.
Denominators are in A051715.
Cf. A027642, A164555.

function a(n,k)
  if n eq 0 then return 1/(k+1);
  else return (k+1)*(a(n-1,k) - a(n-1,k+1));
  end if;
end function;
A051714:= func< n,k | Numerator(a(n,k)) >;
[A051714(k,n-k): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 22 2023

a:= proc(n,k) option remember;
      `if`(n=0, 1/(k+1), (k+1)*(a(n-1,k)-a(n-1,k+1)))
    end:
seq(seq(numer(a(n, d-n)), n=0..d), d=0..12); # Alois P. Heinz, Apr 17 2013

nmax = 12; a[0, k_]:= 1/(k+1); a[n_, k_]:= a[n, k]= (k+1)(a[n-1, k]-a[n-1, k+1]); Numerator[Flatten[Table[a[n-k, k], {n,0,nmax}, {k, n, 0, -1}]]] (* Jean-François Alcover, Nov 28 2011 *)

def a(n,k):
    if (n==0): return 1/(k+1)
    else: return (k+1)*(a(n-1, k) - a(n-1, k+1))
def A051714(n,k): return numerator(a(n, k))
flatten([[A051714(k, n-k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Apr 22 2023

From Fabián Pereyra, Jan 14 2023: (Start)
a(n,k) = numerator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)/(j+k+1)).
E.g.f.: A(x,t) = (x+log(1-t))/(1-t-exp(-x)) = (1+(1/2)*x+(1/6)*x^2/2!-(1/30)*x^4/4!+...)*1 + (1/2+(1/3)*x+(1/6)*x^2/2!+...)*t + (1/3+(1/4)*x+(3/20)*x^2/2!+...)*t^2 + .... (End)

More terms from James Sellers, Dec 07 1999

A051715 Denominators of table a(n,k) read by antidiagonals: a(0,k) = 1/(k+1), a(n+1,k) = (k+1)(a(n,k)-a(n,k+1)), n >= 0, k >= 0.

Original entry on oeis.org

1, 2, 2, 3, 3, 6, 4, 4, 6, 1, 5, 5, 20, 30, 30, 6, 6, 15, 20, 30, 1, 7, 7, 42, 35, 140, 42, 42, 8, 8, 28, 84, 105, 28, 42, 1, 9, 9, 72, 84, 1, 105, 140, 30, 30, 10, 10, 45, 120, 140, 28, 105, 20, 30, 1, 11, 11, 110, 495, 3960, 924, 231, 165, 220, 66, 66, 12, 12, 66, 55, 495, 264, 308, 132, 165, 44, 66, 1

Offset: 0

N. J. A. Sloane

Leading column gives the Bernoulli numbers A027641/A027642.

			Table begins:
    1    1/2   1/3    1/4   1/5  1/6  1/7 ...
   1/2   1/3   1/4    1/5   1/6  1/7 ...
   1/6   1/6   3/20   2/15  5/42 ...
    0    1/30  1/20   2/35  5/84 ...
  -1/30 -1/30 -3/140 -1/105 ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
Index entries for sequences related to Bernoulli numbers.

Rows 2, 3, 4 give A026741/A045896, A051712/A051713, A051722/A051723.
Columns 0, 1, 2, 3 give A000367/A002445, A051716/A051717, A051718/A051719, A051720/A051721.
Numerators are in A051714.

a:= proc(n,k) option remember;
      `if`(n=0, 1/(k+1), (k+1)*(a(n-1,k)-a(n-1,k+1)))
    end:
seq(seq(denom(a(n, d-n)), n=0..d), d=0..12); # Alois P. Heinz, Apr 17 2013

nmax = 12; a[0, k_] := 1/(k+1); a[n_, k_] := a[n, k] = (k+1)(a[n-1, k]-a[n-1, k+1]); Denominator[ Flatten[ Table[ a[n-k, k], {n, 0, nmax}, {k, n, 0, -1}]]](* Jean-François Alcover, Nov 28 2011 *)

a(n,k) = denominator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)/(j+k+1)). - Fabián Pereyra, Jan 14 2023

More terms from James Sellers, Dec 08 1999

A045896 Denominator of n/((n+1)*(n+2)) = A026741/A045896.

Original entry on oeis.org

1, 6, 6, 20, 15, 42, 28, 72, 45, 110, 66, 156, 91, 210, 120, 272, 153, 342, 190, 420, 231, 506, 276, 600, 325, 702, 378, 812, 435, 930, 496, 1056, 561, 1190, 630, 1332, 703, 1482, 780, 1640, 861, 1806, 946, 1980, 1035, 2162, 1128, 2352, 1225, 2550, 1326, 2756, 1431

Offset: 0

Ralf W. Grosse-Kunstleve, Dec 11 1999

Also period length divided by 2 of pairs (a,b), where a has period 2*n-2 and b has period n.
From Paul Curtz, Apr 17 2014: (Start)
Difference table of A026741/A045896:
   0,      1/6,    1/6,    3/20,     2/15,    5/42, ...
   1/6,      0,  -1/60,   -1/60,    -1/70,   -1/84, ... = 1/6, -A051712/A051713
  -1/6,  -1/60,      0,   1/420,    1/420,   1/504, ...
   3/20,  1/60,  1/420,       0,  -1/2520, -1/2520, ...
  -2/15, -1/70, -1/420, -1/2520,        0, 1/13860, ...
   5/42,  1/84,  1/504,  1/2520, -1/13860,       0, ...
Autosequence of the first kind. The main diagonal is A000004. The first two upper diagonals are equal. Their denominators are A000911. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Ralf W. Grosse-Kunstleve, Origin of EIS sequences A045895 & A045896. [Wayback Machine copy]
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), Article 00.2.9.

Factor of A160466.

Cf. A000004, A000384, A000911, A045895, A026741, A051714, A241269.

import Data.Ratio ((%), denominator)
a045896 n = denominator $ n % ((n + 1) * (n + 2))
-- Reinhard Zumkeller, Dec 12 2011

seq((n+1)*(n+2)*(3-(-1)^n)/4, n=0..20); # C. Ronaldo
with(combinat): seq(lcm(n+1,binomial(n+2,n)), n=0..50); # Zerinvary Lajos, Apr 20 2008

Table[LCM[2*n + 2, n + 2]/2, {n, 0, 40}] (* corrected by Amiram Eldar, Sep 14 2022 *)
Denominator[#[[1]]/(#[[2]]#[[3]])&/@Partition[Range[0,60],3,1]] (* Harvey P. Dale, Aug 15 2013 *)

Vec((2*x^3+3*x^2+6*x+1)/(1-x^2)^3+O(x^99)) \\ Charles R Greathouse IV, Mar 23 2016

G.f.: (2*x^3+3*x^2+6*x+1)/(1-x^2)^3.
a(n) = (n+1)*(n+2) if n odd; or (n+1)*(n+2)/2 if n even = (n+1)*(n+2)*(3-(-1)^n)/4. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004
a(2*n) = A000384(n+1); a(2*n+1) = A026741(n+1). - Reinhard Zumkeller, Dec 12 2011
Sum_{n>=0} 1/a(n) = 1 + log(2). - Amiram Eldar, Sep 11 2022
From Amiram Eldar, Sep 14 2022: (Start)
a(n) = lcm(2*n+2, n+2)/2.
a(n) = A045895(n+2)/2. (End)
E.g.f.: (2 + 8*x + x^2)*cosh(x)/2 + (2 + 2*x + x^2)*sinh(x). - Stefano Spezia, Apr 24 2024

A051713 Denominator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/A045896.

Original entry on oeis.org

1, 60, 60, 70, 84, 504, 120, 990, 165, 572, 1092, 2730, 280, 4080, 2448, 1938, 855, 7980, 1540, 10626, 3036, 4600, 7800, 17550, 819, 21924, 12180, 8990, 7440, 32736, 5984, 39270, 5355, 15540, 25308, 54834, 4940, 63960, 34440

Offset: 1

N. J. A. Sloane

			0, 1/60, 1/60, 1/70, 1/84, 5/504, 1/120, 7/990, 1/165, 3/572,...

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Cf. A051712. Row 3 of table in A051714/A051715.

Denominator[#[[1]]-#[[2]]&/@(Partition[#[[1]]/(#[[2]]#[[3]])&/@Partition[ Range[50],3,1],2,1])] (* Harvey P. Dale, Nov 15 2014 *)

cp's OEIS Frontend

A164848 a(n) = A026741(n)/A051712(n+1).

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A051714 Numerators of table a(n,k) read by antidiagonals: a(0,k) = 1/(k+1), a(n+1,k) = (k+1)*(a(n,k) - a(n,k+1)), n >= 0, k >= 0.

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A051715 Denominators of table a(n,k) read by antidiagonals: a(0,k) = 1/(k+1), a(n+1,k) = (k+1)(a(n,k)-a(n,k+1)), n >= 0, k >= 0.

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A045896 Denominator of n/((n+1)*(n+2)) = A026741/A045896.

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A051713 Denominator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/A045896.

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