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A238157
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Reduced denominators of integral of the Stirling numbers of first kind.
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1
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1, 1, 2, 1, 2, 3, 1, 1, 1, 4, 1, 1, 3, 2, 5, 1, 1, 3, 4, 1, 6, 1, 1, 3, 4, 1, 2, 7, 1, 1, 1, 1, 1, 6, 1, 8, 1, 1, 1, 1, 5, 3, 1, 2, 9, 1, 1, 1, 1, 5, 2, 1, 4, 1, 10, 1, 1, 1, 1, 1, 2, 1, 4, 3, 2, 11, 1, 1, 1, 1, 1, 3, 1, 8, 3, 1, 1, 12
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OFFSET
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0,3
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COMMENTS
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s(n,k), signed Stirling numbers of the first kind (A048994):
1,
0, 1,
0, -1, 1,
0, 2, -3, 1,
0, -6, 11, -6, 1
etc.
The unsigned numbers, abs(s(n,k)), are the unsigned Stirling numbers of the first kind, A132393(n).
For the integration of these triangles we divide by A002260(n+1). For the first one the reduced numbers are
1,
0, 1/2,
0, -1/2, 1/3,
0, 1, -1, 1/4,
0, -3, 11/3, -3/2, 1/5,
etc.
Hence the denominators in the example.
Because the integration is between 0 and 1, the fractions appear in a numerical Adams integration with the denominators multiplied by n!, i.e., 1, 1/2, -1/12, 1/24, -19/720, 27/1440, ... . Reference, array p. 36.
(*) The Cauchy numbers of the first type or the Bernoulli numbers of the second kind.
Without signs, the row sums are 1, 1/2, 5/6, 9/4, 251/30, 475/12, ... = A002657(n)/A002790(n), Cauchy numbers of the second type. See Nørlund numbers, 1924.
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REFERENCES
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P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969 (see array p. 56).
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924
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LINKS
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FORMULA
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EXAMPLE
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Denominators triangle (a(n)):
1,
1, 2
1, 2, 3,
1, 1, 1, 4,
1, 1, 3, 2, 5,
1, 1, 3, 4, 1, 6,
1, 1, 3, 4, 1, 2, 7,
etc.
The Least Common Multiples are A002790. The second column is A141044(n).
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MATHEMATICA
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Table[StirlingS1[n, k]/(k+1) // Denominator, {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2014 *)
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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