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A190339
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The denominators of the subdiagonal in the difference table of the Bernoulli numbers.
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26
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2, 6, 15, 105, 105, 231, 15015, 2145, 36465, 969969, 4849845, 10140585, 10140585, 22287, 3231615, 7713865005, 7713865005, 90751353, 218257003965, 1641030105, 67282234305, 368217318651, 1841086593255
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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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(n, n) = 2*T(n, n+1).
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EXAMPLE
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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,
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MAPLE
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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:
denom( T(n+1, n)) ;
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MATHEMATICA
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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 *)
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PROG
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(Sage)
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)]
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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