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A006569
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Numerators of generalized Bernoulli numbers.
(Formerly M3731)
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3
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1, -1, 1, 1, -1, -5, -1, 7, 13, -307, -479, 1837, 100921, 15587, -23737, -5729723, 14731223, 9129833, 2722952839, -4700745901, -1556262845, 190717213397, 24684889339847, -50242799489, -148437433077277, -8592042383621, 221844330989749, 176585172615885307, -9245931549625447
(list;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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0,6
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REFERENCES
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F. T. Howard, A sequence of numbers related to the exponential function, Duke Math. J., 34 (1967), 599-615.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Recurrence relation: Bernoulli(n+1) = a(n+1) - Sum_{r=1..n+1} binomial(n+1, r)*Bernoulli(r)*a(n+2-r); a(0)=1 (p. 603 of the Howard reference). - Emeric Deutsch, Jan 23 2005
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MAPLE
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eq:=n->bernoulli(n+1)=a[n+1]-sum(binomial(n+1, r)*bernoulli(r)*a[n+2-r], r=1..n+1): a[0]:=1:for n from 0 to 28 do a[n+1]:=solve(eq(n), a[n+1]) od: seq(numer(a[n]), n=0..29); # Emeric Deutsch, Jan 23 2005
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MATHEMATICA
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rows = 29; M = Table[If[n-1 <= k <= n, 0, Binomial[n, k]], {n, 2, rows+1}, {k, 0, rows-1}] // Inverse;
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PROG
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(Sage)
f, R, C = 1, [1], [1]+[0]*(len-1)
for n in (1..len-1):
f *= n
for k in range(n, 0, -1):
C[k] = C[k-1] / (k+2)
C[0] = -sum(C[k] for k in (1..n))
R.append((C[0]*f).numerator())
return R
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CROSSREFS
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KEYWORD
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frac,sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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