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A085737
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Numerators in triangle formed from Bernoulli numbers.
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9
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1, 1, 1, 1, 1, 1, 0, 1, 1, 0, -1, 1, 2, 1, -1, 0, -1, 1, 1, -1, 0, 1, -1, -1, 8, -1, -1, 1, 0, 1, -1, 4, 4, -1, 1, 0, -1, 1, -1, -4, 8, -4, -1, 1, -1, 0, -1, 1, -8, 4, 4, -8, 1, -1, 0, 5, -5, 7, 4, -116, 32, -116, 4, 7, -5, 5, 0, 5, -5, 32, -28, 16, 16, -28, 32, -5, 5, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,13
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COMMENTS
| Triangle is determined by rules 0) the top number is 1; 1) each number is the sum of the two below it; 2) it is left-right symmetric; 3) the numbers in each of the border rows, after the first 3, are alternately 0.
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REFERENCES
| Lange, Fabien; and Grabisch, Michel; The interaction transform for functions on lattices. Discrete Math. 309 (2009), no. 12, 4037-4048. [From N. J. A. Sloane, Nov 26 2011]
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FORMULA
| T(n, 0) = (-1)^n*Bernoulli(n), T(n, k) = T(n-1, k-1) - T(n, k-1) for k=1..n.
T(n,k) = Sum_{j=0..k} binomial(k,j)*Bernoulli(n-j). [Lange and Grabisch]
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EXAMPLE
| Triangle of fractions begins
1
1/2, 1/2
1/6, 1/3, 1/6
0, 1/6, 1/6, 0
-1/30, 1/30, 2/15, 1/30, -1/30
0, -1/30, 1/15, 1/15, -1/30, 0
1/42, -1/42, -1/105, 8/105, -1/105, -1/42, 1/42
0, 1/42, -1/21, 4/105, 4/105, -1/21, 1/42, 0
-1/30, 1/30, -1/105, -4/105, 8/105, -4/105, -1/105, 1/30, -1/30
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MAPLE
| nmax:=11: for n from 0 to nmax do T(n, 0):= (-1)^n*bernoulli(n) od: for n from 1 to nmax do for k from 1 to n do T(n, k) := T(n-1, k-1) - T(n, k-1) od: od: for n from 0 to nmax do seq(T(n, k), k=0..n) od; Tx:=0: for n from 0 to nmax do for k from 0 to n do A085737(Tx):= numer(T(n, k)): Tx:=Tx+1: od: od: seq(A085737(n), n=0..Tx-1); [Johannes W. Meijer, Jun 29 2011]
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CROSSREFS
| Cf. A085738. See A051714/A051715 for another triangle that generates the Bernoulli numbers.
Sequence in context: A037800 A144411 A138253 * A191904 A005090 A073490
Adjacent sequences: A085734 A085735 A085736 * A085738 A085739 A085740
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KEYWORD
| sign,frac
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), following a suggestion of J. H. Conway, Jul 23 2003
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EXTENSIONS
| Sign flipped in formula by Johannes W. Meijer, Jun 29 2011
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