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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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8
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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 formulae.
Reading the array T(n, m), see the examples, by its antidiagonals leads to A085737(n)/A085738(n).
A164555(n)/A027642(n) is an eigensequence (its inverse binomial transform is the sequence signed) of the second kind; The main diagonal T(n, n) is double 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
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REFERENCES
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Ludwig Seidel, Ueber 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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Table of n, a(n) for n=0..22.
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
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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(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
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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:
A190339 := proc(n)
denom( T(n+1, n)) ;
end proc: # R. J. Mathar, Apr 25 2013
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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)
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
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CROSSREFS
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Sequence in context: A216811 A009455 A007709 * A078328 A038111 A181993
Adjacent sequences: A190336 A190337 A190338 * A190340 A190341 A190342
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KEYWORD
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nonn,frac,changed
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AUTHOR
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Paul Curtz, May 09 2011
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EXTENSIONS
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Edited and Maple program added by Johannes W. Meijer, Jun 29 2011, Jun 30 2011.
New name by Peter Luschny, Jun 21 2012.
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STATUS
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approved
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