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A240581
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Array read by antidiagonals: numerators of the core of the classical Bernoulli numbers.
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3
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2, -1, 1, -1, -8, -1, 1, 4, -4, -1, -1, 4, 8, 4, -1, -1, -8, -4, 4, 8, 1, 7, -4, -116, -32, -116, -4, 7, 5, 32, 28, 16, -16, -28, -32, -5, -2663, -388, 2524, 5072, 6112, 5072, 2524, -388, -2663, -691, -10264, -10652, -8128, -3056, 3056, 8128, 10652, 10264, 691
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OFFSET
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0,1
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COMMENTS
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Sum of antidiagonals: 2/15, 0, -2/21, 0, 2/15, 0, -10/33, 0, 1382/1365,... =-4*A164555(n+4)/A027642(n+4).
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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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EXAMPLE
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As a triangle:
2,
-1, 1,
-1, -8, -1,
1, 4, -4, -1,
-1, 4, 8, 4, -1,
etc.
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MAPLE
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DifferenceTableBernoulli := proc(n) local A, m, k; A := array(0..n, 0..n);
# pritty print
for m from 0 to n do for k from 0 to n do A[m, k] := '~' od od;
# compute elements
for m from 0 to n do A[m, 0] := bernoulli(m, 1);
for k from m-1 by -1 to 0 do
A[k, m-k] := A[k+1, m-k-1] - A[k, m-k-1] od od;
convert(A, matrix) end:
A := DifferenceTableBernoulli(13); L := NULL;
for n from 0 to 9 do for k from 0 to n do
L := L, numer(A[3+k, 3+n-k]) od od;
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MATHEMATICA
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max = 13; tb = Table[BernoulliB[n], {n, 0, max}]; td = Table[Differences[tb, n][[3 ;; -1]], {n, 2, max - 1}]; Table[td[[n - k + 1, k]] // Numerator, {n, 1, max - 3}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 11 2014 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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