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A006233
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Denominators of Cauchy numbers of first type.
(Formerly M1558)
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37
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1, 2, 6, 4, 30, 4, 84, 24, 90, 20, 132, 8, 5460, 840, 360, 48, 1530, 4, 1596, 168, 1980, 1320, 8280, 80, 81900, 6552, 1512, 112, 3480, 80, 114576, 7392, 117810, 7140, 1260, 8, 3838380, 5928, 936, 48, 81180, 440, 1191960, 55440, 869400, 38640, 236880, 224
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OFFSET
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0,2
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COMMENTS
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The corresponding numerators are given in A006232.
The signed rationals A006232(n)/a(n) provide the a-sequence for the Stirling2 Sheffer matrix A048993. See the W. Lang link concerning Sheffer a- and z-sequences.
Cauchy numbers of the first type are also called Bernoulli numbers of the second kind.
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics, Cambridge, 1946, p. 259.
L. Jolley, Summation of Series, Chapman and Hall, London, 1925, pp. 14-15 (formula 70).
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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Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math. 306 (2006), no. 16, 1906-1920.
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FORMULA
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Denominator of integral of x(x-1)...(x-n+1) from 0 to 1.
E.g.f.: x/log(1+x).
a(n) = denominator(f(n) * n!), where f(0) = 1, f(n) = Sum_{k=0..n-1} (-1)^(n-k+1) * f(k) / (n-k+1). - Daniel Suteu, Feb 23 2018
Sum_{k = 1..n} (1/k) = A001620 + log(n) + 1/(2*n) - Sum_{k >= 2} abs((A006232(k)/a(k)/k/(Product_{j = 0..k-1} (n-j)))), (see I. S. Gradsteyn, I. M. Ryzhik). - A.H.M. Smeets, Nov 14 2018
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EXAMPLE
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1, 1/2, -1/6, 1/4, -19/30, 9/4, -863/84, 1375/24, -33953/90,...
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MAPLE
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seq(denom(add(stirling1(n, k)/(k+1), k=0..n)), n=0..12); # Peter Luschny, Apr 28 2009
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MATHEMATICA
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With[{nn=50}, Denominator[CoefficientList[Series[x/Log[1+x], {x, 0, nn}], x] Range[0, nn]!]] (* Harvey P. Dale, Oct 28 2011 *)
Join[{1}, Array[Abs@Denominator[ Integrate[Product[(x - k), {k, 0, # - 1}], {x, 0, 1}]] &, 50]] (* Michael De Vlieger, Nov 13 2018 *)
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PROG
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(PARI) for(n=0, 50, print1(denominator( sum(k=0, n, stirling(n, k, 1)/(k+1)) ), ", ")) \\ G. C. Greubel, Nov 13 2018
(Magma) [Denominator((&+[StirlingFirst(n, k)/(k+1): k in [0..n]])): n in [0..50]]; // G. C. Greubel, Nov 13 2018
(Sage)
f, R, C = 1, [1], [1]+[0]*(len-1)
for n in (1..len-1):
for k in range(n, 0, -1):
C[k] = -C[k-1] * k / (k + 1)
C[0] = -sum(C[k] for k in (1..n))
R.append((C[0]*f).denominator())
f *= n+1
return R
(Python) # Results are abs values
from fractions import gcd
aa, n, sden = [0, 1], 1, 1
print(0, 1)
while n < 20:
j, snom, sden, a = 1, 0, (n+1)*sden, 0
while j < len(aa):
snom, j = snom+aa[j]*(sden//(j+1)), j+1
nom, den = snom, sden
print(n, den//gcd(nom, den))
aa, j = aa+[-aa[j-1]], j-1
while j > 0:
aa[j], j = n*aa[j]-aa[j-1], j-1
(Python)
from fractions import Fraction
from sympy.functions.combinatorial.numbers import stirling
def A006233(n): return sum(Fraction(stirling(n, k, kind=1, signed=True), k+1) for k in range(n+1)).denominator # Chai Wah Wu, Jul 09 2023
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CROSSREFS
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KEYWORD
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nonn,frac,nice,easy
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AUTHOR
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STATUS
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approved
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