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A238157 Reduced denominators of integral of the Stirling numbers of first kind.

%I A238157 #17 Dec 21 2014 05:09:49
%S A238157 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,
%T A238157 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,
%U A238157 1,1,1,1,3,1,8,3,1,1,12
%N A238157 Reduced denominators of integral of the Stirling numbers of first kind.
%C A238157 s(n,k), signed Stirling numbers of the first kind (A048994):
%C A238157 1,
%C A238157 0,  1,
%C A238157 0, -1,   1,
%C A238157 0,  2,  -3,  1,
%C A238157 0, -6,  11, -6, 1
%C A238157 etc.
%C A238157 The unsigned numbers, abs(s(n,k)), are the unsigned Stirling numbers of the first kind, A132393(n).
%C A238157 For the integration of these triangles we divide by A002260(n+1). For the first one the reduced numbers are
%C A238157 1,
%C A238157 0,  1/2,
%C A238157 0, -1/2,  1/3,
%C A238157 0,    1,   -1,  1/4,
%C A238157 0,   -3, 11/3, -3/2, 1/5,
%C A238157 etc.
%C A238157 Hence the denominators in the example.
%C A238157 Sums by rows: 1, 1/2, -1/6, 1/4, -19/30, 27/12 = 9/4, = (-1)^(n+1)*A141417(n)/A002790(n) = A006232(n)/A006233(n) (*).
%C A238157 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.
%C A238157 (*) The Cauchy numbers of the first type or the Bernoulli numbers of the second kind.
%C A238157 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.
%D A238157 P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969 (see array p. 56).
%D A238157 N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924
%F A238157 Denominators of reduced A132393(n)/A002260(n+1).
%e A238157 Denominators triangle (a(n)):
%e A238157 1,
%e A238157 1, 2
%e A238157 1, 2, 3,
%e A238157 1, 1, 1, 4,
%e A238157 1, 1, 3, 2, 5,
%e A238157 1, 1, 3, 4, 1, 6,
%e A238157 1, 1, 3, 4, 1, 2, 7,
%e A238157 etc.
%e A238157 The Least Common Multiples are A002790. The second column is A141044(n).
%t A238157 Table[StirlingS1[n, k]/(k+1) // Denominator, {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 21 2014 *)
%Y A238157 Cf. A091137.
%K A238157 nonn,frac
%O A238157 0,3
%A A238157 _Paul Curtz_, Feb 18 2014