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

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

Offset: 0

Author

Paul Curtz, Feb 18 2014

Keywords

• nonn
• frac

Comments

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.

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) (*).

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.

Examples

			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).
		

References

• 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

Crossrefs

Cf. A091137.

Programs

Mathematica

Table[StirlingS1[n, k]/(k+1) // Denominator, {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2014 *)

Formula

Denominators of reduced A132393(n)/A002260(n+1).