A051717 -id:A051717 - cp's OEIS frontend

A140812 a(n) = A051717(2n) + A051717(2n+1).

3, 9, 60, 84, 60, 132, 5460, 12, 1020, 1596, 660, 276, 5460, 12, 1740, 28644, 1020, 12, 3838380, 12, 27060, 3612, 1380, 564, 92820, 132, 3180, 1596, 1740, 708, 113573460, 12, 1020, 129444, 60, 9372, 280201740, 12, 60, 6636, 460020, 996, 6808620, 12, 122820

Offset: 0

Paul Curtz, Jul 16 2008

nonn

All terms are multiples of 3.

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A129826, A140333.

C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end:
A051717 := proc(n) denom(C(n)) ; end: A140812 := proc(n) A051717(2*n)+A051717(2*n+1) ; end: seq(A140812(n),n=0..80) ; # R. J. Mathar, Jun 28 2009

A051717 := Join[{1}, Denominator[Total /@ Partition[BernoulliB[Range[0, 500]], 2, 1]]]; Join[{3, 9}, Table[2*A051717[[2*n]], {n, 3,30}]] (* G. C. Greubel, Dec 22 2017 *)

Edited and extended by R. J. Mathar, Jun 28 2009

A132084 A051717(2n).

Original entry on oeis.org

1, 3, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, 1806, 690, 282, 46410, 66, 1590, 798, 870, 354, 56786730, 6, 510, 64722, 30, 4686, 140100870, 6, 30, 3318, 230010, 498, 3404310, 6, 61410, 272118, 1410, 6, 4501770

Offset: 0

Paul Curtz, Aug 26 2008

Essentially the same as A006954.

a(2n) + a(2n+1) = 4, 72, 96, 2736, 1308, 468, ... are multiples of 4.

C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end;
A051717 := proc(n) denom(C(n)) ; end: A132084 := proc(n) A051717(2*n) ; end: seq(A132084(n),n=0..120) ; # R. J. Mathar, Sep 07 2009

Extended by R. J. Mathar, Sep 07 2009

A153087 a(n) = A051717(3n) + A051717(3n+1) + A051717(3n+2).

Original entry on oeis.org

6, 66, 114, 162, 5466, 1026, 1926, 606, 5466, 1746, 29154, 522, 3838386, 27066, 4302, 1254, 92886, 3246, 2466, 1578, 113573466, 1026, 129474, 9402, 280201746, 66, 236646, 231006, 6808626, 122826, 545646, 1422, 9003546, 66666, 10242, 2874, 418384938, 3344058

Offset: 0

Paul Curtz, Dec 18 2008

nonn

All terms are multiples of 6.

The first differences are 60, 48, 48, 5304, -4440, ... Apparently (checked for the first 700 entries) these are multiples of 12.

Cf. A051717, A140812.

C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end:
A051717 := proc(n) denom(C(n)) ; end: A153087 := proc(n) A051717(3*n)+A051717(3*n+1)+A051717(3*n+2) ; end: seq(A153087(n),n=0..120) ; # R. J. Mathar, Sep 07 2009

Extended by R. J. Mathar, Sep 07 2009

A172086 Numerators of sum (C(n) = A051716/A051717) + (1 followed by first differences A172083/A051717 of Bernoulli numbers).

Original entry on oeis.org

2, -2, 1, -1, -1, 1, 1, -1, -1, 1, 5, -5, -691, 691, 7, -7, -3617, 3617, 43867, -43867, -174611, 174611, 854513, -854513, -236364091, 236364091, 8553103, -8553103, -23749461029, 23749461029, 8615841276005, -8615841276005, -7709321041217, 7709321041217, 2577687858367, -2577687858367

Offset: 0

Paul Curtz, Jan 25 2010

Denominators: 1, 1, 3, 3, 15, 15, 21, 21, 15, 15, 33, 33, 1365, 1365, ... = A001897 with terms repeated. See A000367/A002445.

Cf. A000367, A001897, A002445, A027641, A051716, A051717

Name edited by Michel Marcus, Jan 30 2021

Clarified definition, added more terms. - N. J. A. Sloane, Apr 22 2021

A176511 From Bernoulli twin numbers to Catalan numbers arrays (*).First part.We consider array, from Bernoulli twin numbers A051716/A051717 mixed with their companion A172083/A051717 BTC(n)=1,1,-1/2,-3/2,-1/3,2/3,-1/6,-1/6, and successive differences ,named BTC1. a(n) are numerators of BTC(n).Denominators are (double A051717)=1,1,2,2,3,3,6,6,30,30,30,30,.

Original entry on oeis.org

1, 1, -1, -3, -1, 2, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 5, 5

Offset: 0

Paul Curtz, Apr 19 2010

(*) Even case:ECT(n) in A176239. BTC(n) are not Bi-Bernoulli numbers (absolute values of mixed sequences are not the same like BB1(n) in A176144 or BB2(n) in A176184). Rows of array BTC1: 1) 1,1,-1/2,-3/2,-1/3,2/3,-1/6,-1/6,-1/30,-1/30,1/30,1/30,1/42,1/42,; 2) 0,-3/2,-1,7/6,1,-5/6,0,2/15,0,2/15,0; 3) -3/2,1/2,13/6,-1/6,-11/6,5/6,2/15,-2/15,2/15,-2/15; 4) 2,5/3,-7/3,-5/3,8/3,-7/10,-4/15,4/15,-4/15; 5) -1/3,-4,2/3,13/3,-101/30,13/30,8/15,-8/15; 6) -11/3,14/3,11/3,-77/10,19/5,1/10,-16/15; 7) 25/3,-1, -341/30,23/2,-37/10,-29/30; 8) -28/3,-311/30,343/15,. Correction:in A176150 last term (-517) is false.

A176784 From Bernoulli twin numbers to Catalan numbers arrays.Second part.Consider array from companion of Bernoulli twin numbers A172083/A051717 mixed with A051716/A051717 BCT(n)=1,1,-3/2,-1/2,2/3,-1/3,-1/6,-1/6, with successive differences,named BCT1.a(n) are numerators of BCT(n).Denominators (double A051717)=1,1,2,2,3,3,6,6,30,30,30,30,.

Original entry on oeis.org

1, 1, -3, -1, 2, -1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 5, 5

Offset: 0

Paul Curtz, Apr 26 2010

a(n) is A176511 (companion) with A176511(2),A176511(3), A176511(4),A176511(5) swapped by pairs.Rows of BCT1: 1) 1,1,-3/2,-1/2,2/3,-1/3,-1/6,-1/6; 2) 0,-5/2,1,7/6,-1,1/6,0,2/15; 3) -5/2,7/2,1/6,-13/6,7/6,-1/6,2/15,-2/15; 4) 6,-10/3,-7/3,10/3,-4/3,3/10,-4/15,4/15; 5) -28/3,1,17/3,-14/3,49/30,-17/30,8/15,-8/15; 6) 31/3,14/3,-31/3,63/10,-11/5,11/10,16/15; 7) -17/3,-15,499/30,-17/2,33/10,-1/30; 8) -28/3,949/30,-377/15; .Now we subtract first part BTC1 and second BCT1.Hence an array with only integers.We consider it from seventh column from right to left.Columns changed into rows give different possibilities for Catalan numbers A000108 or A000108(n+1). Among them,ECT(n) in A176239. Odd triangle is 1, 1,0,-1, 0,1,-1,0,2, 0,0,1,-2,2,0,-5, 0,0,0,1,-3,5,-5,0,14, .

A190339 The denominators of the subdiagonal in the difference table of the Bernoulli numbers.

Original entry on oeis.org

2, 6, 15, 105, 105, 231, 15015, 2145, 36465, 969969, 4849845, 10140585, 10140585, 22287, 3231615, 7713865005, 7713865005, 90751353, 218257003965, 1641030105, 67282234305, 368217318651, 1841086593255

Offset: 0

Paul Curtz, May 09 2011

Apparently a(n) = A181131(n) for n>=2 (checked numerically up to n=640). - R. J. Mathar, Aug 25 2025
The denominators of the T(n, n+1) with T(0, m) = A164555(m)/A027642(m) and T(n, m) = T(n-1, m+1) - T(n-1, m), n >= 1, m >= 0. For the numerators of the T(n, n+1) see A191972.
The T(n, m) are defined by A164555(n)/A027642(n) and its successive differences, see the formulas.
Reading the array T(n, m), see the examples, by its antidiagonals leads to A085737(n)/A085738(n).
A164555(n)/A027642(n) is an autosequence (eigensequence whose inverse binomial transform is the sequence signed) of the second kind; the main diagonal T(n, n) is twice the first upper diagonal T(n, n+1).
We can get the Bernoulli numbers from the T(n, n+1) in an original way, see A192456/A191302.
Also the denominators of T(n, n+1) of the table defined by A085737(n)/A085738(n), the upper diagonal, called the median Bernoulli numbers by Chen. As such, Chen proved that a(n) is even only for n=0 and n=1 and that a(n) are squarefree numbers. (see Chen link). - Michel Marcus, Feb 01 2013
The sum of the antidiagonals of T(n,m) is 1 in the first antidiagonal, otherwise 0. Paul Curtz, Feb 03 2015

			The first few rows of the T(n, m) array (difference table of the Bernoulli numbers) are:
1,       1/2,     1/6,      0,     -1/30,         0,        1/42,
-1/2,   -1/3,    -1/6,  -1/30,      1/30,      1/42,       -1/42,
1/6,     1/6,    2/15,   1/15,    -1/105,     -1/21,      -1/105,
0,     -1/30,   -1/15, -8/105,    -4/105,     4/105,       8/105,
-1/30, -1/30,  -1/105,  4/105,     8/105,     4/105,   -116/1155,
0,      1/42,    1/21,  4/105,    -4/105,   -32/231,     -16/231,
1/42,   1/42,  -1/105, -8/105, -116/1155,    16/231,  6112/15015,

Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

G. C. Greubel, Table of n, a(n) for n = 0..450
Kwang-Wu Chen, A summation on Bernoulli numbers, Journal of Number Theory, Volume 111, Issue 2, April 2005, Pages 372-391.
Peter Luschny, Computation and asymptotics of the Bernoulli numbers

T := proc(n,m)
    option remember;
    if n < 0 or m < 0 then
        0 ;
    elif n = 0 then
        if m = 1 then
            -bernoulli(m) ;
        else
            bernoulli(m) ;
        end if;
    else
        procname(n-1,m+1)-procname(n-1,m) ;
    end if;
end proc:
A190339 := proc(n)
    denom( T(n+1,n)) ;
end proc: # R. J. Mathar, Apr 25 2013

nmax = 23; b[n_] := BernoulliB[n]; b[1]=1/2; bb = Table[b[n], {n, 0, 2*nmax-1}]; diff = Table[Differences[bb, n], {n, 1, nmax}]; Diagonal[diff] // Denominator (* Jean-François Alcover, Aug 08 2012 *)

def A190339_list(n) :
    T = matrix(QQ, 2*n+1)
    for m in (0..2*n) :
        T[0,m] = bernoulli_polynomial(1,m)
        for k in range(m-1,-1,-1) :
            T[m-k,k] = T[m-k-1,k+1] - T[m-k-1,k]
    for m in (0..n-1) : print([T[m,k] for k in (0..n-1)])
    return [denominator(T[k,k+1]) for k in (0..n-1)]
A190339_list(7) # Also prints the table as displayed in EXAMPLE. Peter Luschny, Jun 21 2012

T(0, m) = A164555(m)/A027642(m) and T(n, m) = T(n-1, m+1) - T(n-1, m), n >= 1, m >= 0.
T(1, m) = A051716(m+1)/A051717(m+1);
T(n, n) = 2*T(n, n+1).
T(n+1, n+1) = (-1)^(1+n)*A181130(n+1)/A181131(n+1). - R. J. Mathar, Jun 18 2011
a(n) = A141044(n)*A181131(n). - Paul Curtz, Apr 21 2013

Edited and Maple program added by Johannes W. Meijer, Jun 29 2011, Jun 30 2011

New name from Peter Luschny, Jun 21 2012

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

A051716 Numerators of Bernoulli twin numbers C(n).

Original entry on oeis.org

1, -1, -1, -1, -1, 1, 1, -1, -1, 1, 5, -5, -691, 691, 7, -7, -3617, 3617, 43867, -43867, -174611, 174611, 854513, -854513, -236364091, 236364091, 8553103, -8553103, -23749461029, 23749461029, 8615841276005, -8615841276005, -7709321041217, 7709321041217, 2577687858367

Offset: 0

N. J. A. Sloane

The Bernoulli twin numbers C(n) are defined by C(0) = 1, then C(2n) = B(2n) + B(2n-1), C(2n+1) = -B(2n+1) - B(2n), where B() are the Bernoulli numbers A027641/A027642. The definition is due to Paul Curtz.
For denominators see A051717.
Negatives of numerators of column 1 of table described in A051714/A051715.

			The C(n) sequence is 1, -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66, -691/2730, 691/2730, 7/6, -7/6, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..640
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Cf. A000367, A027641, A027642, A051714, A051715.

Cf. A051717, A129825, A129826, A129724, A164555.

f:= func< n | Bernoulli(n) + Bernoulli(n-1) >;
function A051716(n)
  if n eq 0 then return 1;
  elif (n mod 2) eq 0 then return Numerator(f(n));
  else return Numerator(-f(n));
  end if;
end function;
[A051716(n): n in [0..50]]; // G. C. Greubel, Apr 22 2023

C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end;

c[0]= 1; c[n_?EvenQ]:= BernoulliB[n] + BernoulliB[n-1]; c[n_?OddQ]:= -BernoulliB[n] - BernoulliB[n-1]; Table[Numerator[c[n]], {n,0,34}] (* Jean-François Alcover, Dec 19 2011 *)

a(n) = if (n==0, 1, nu = numerator(bernfrac(n)+bernfrac(n-1)); if (n%2, -nu, nu)); \\ Michel Marcus, Jan 29 2017

def f(n): return bernoulli(n)+bernoulli(n-1)
def A051716(n):
    if (n==0): return 1
    elif (n%2==0): return numerator(f(n))
    else: return numerator(-f(n))
[A051716(n) for n in range(51)] # G. C. Greubel, Apr 22 2023

Numerators of differences of the sequence of rational numbers 0 followed by A164555/A027642. - Paul Curtz, Jan 29 2017

The e.g.f. of the rationals a(n)/A051717(n) is -(1/x + x^2/2 + x/(1 - exp(x)) + dilog(exp(-x))), (with dilog(x) = polylog(2, 1-x)). From integrating the e.g.f. of the z-sequence (exp(x) - (1+x))/(exp(x) -1)^2 for the Bernoulli polynomials of the second kind (A290317 / A290318). - Wolfdieter Lang, Aug 07 2017

More terms from James Sellers, Dec 08 1999

Edited by N. J. A. Sloane, May 25 2008

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

cp's OEIS Frontend

A140812 a(n) = A051717(2n) + A051717(2n+1).

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A132084 A051717(2n).

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A153087 a(n) = A051717(3n) + A051717(3n+1) + A051717(3n+2).

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A172086 Numerators of sum (C(n) = A051716/A051717) + (1 followed by first differences A172083/A051717 of Bernoulli numbers).

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A190339 The denominators of the subdiagonal in the difference table of the Bernoulli numbers.

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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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A051716 Numerators of Bernoulli twin numbers C(n).

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