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A238800
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Unreduced numerators in triangle that leads to the Euler numbers A198631(n)/A006519(n+1).
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1
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1, 1, 1, -2, 1, -3, 1, -4, 2, 1, -5, 5, 1, -6, 9, -10, 1, -7, 14, -35, 1, -8, 20, -80, 26, 1, -9, 27, -150, 117, 1, -10, 35, -250, 325, -454, 1, -11, 44, -385, 715, -2497, 1, -12, 54, -560, 1365, -8172, 5914, 1, -13
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OFFSET
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0,4
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COMMENTS
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We use the array ASPEC mentioned in A191302:
2, 1, 1, 1, 1, 1, 1, 1,...
2, 3, 4, 5, 6, 7, 8, 9,...
2, 5, 9, 14, 20, 27, 35, 44,...
2, 7, 16, 30, 50, 77, 112, 156,...
with the first upper diagonal of the difference table of the autosequence A198631(n)/A006519(n+1), i.e., 1/2, -1/4, 1/4, -5/8, 13/4, -227/8, 2957/8,...
written by columns:
1/2
1/2,
1/2, -1/4,
1/2, -1/4,
1/2, -1/4, 1/4,
1/2, -1/4, 1/4,
1/2, -1/4, 1/4, -5/8,
1/2, -1/4, 1/4, -5/8,
etc.
Hence, by multiplication of this double triangle by ASPEC, the beginning of the double triangle ESPEC is obtained:
E(0) = 1 = 1
E(1) = 1/2 = 1/2
E(2) = 0 = 1/2 -2/4
E(3) = -1/4 = 1/2 -3/4
E(4) = 0 = 1/2 -4/4 +2/4
E(5) = 1/2 = 1/2 -5/4 +5/4
E(6) = 0 = 1/2 -6/4 +9/4 -10/8
E(7) = -17/8 = 1/2 -7/4 +14/4 -35/8
E(8) = 0 = 1/2 -8/4 +20/4 -80/8 +26/4.
The terms of the sequence are the reduced numerators. Like A192456(n) for Bernoulli numbers A164555(n)/A027642(n).
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LINKS
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EXAMPLE
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a(n) by triangle
1,
1,
1, -2,
1, -3,
1, -4, 2,
1, -5, 5,
1, -6, 9, -10,
1, -7, 14, -35,
1, -8, 20, -80, 26,
etc.
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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