A141417 -id:A141417 - cp's OEIS frontend

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

Paul Curtz, Feb 18 2014

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.

			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).

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

Cf. A091137.

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

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

A235936 Triangle of numerators of the unreduced coefficients of a numerical integration for a prediction Adams method.

Original entry on oeis.org

1, 1, 1, 5, 8, -1, 9, 19, -5, 1, 251, 646, -264, 106, -19, 475, 1427, -798, 482, -173, 27, 19087, 65112, -46461, 37504, -20211, 6312, -863, 36799, 139849, -121797, 123133, -88547, 41499, -11351, 1375

Offset: 0

Paul Curtz, Jan 17 2014

The coefficients b(q,j) are such that:
(q-j)!*j!*b(q,j) = (-1)^(q-j)*Int (from 0 to 1) u*(u-1)*...*(u-q) du/(u-j).
0<=j<=q, 0<=q<=p where p is the degree (or order) of the numerical integration.
This is the first case of tridimensional b(i,q,j), the integration is from i to i+1, with i=0.
The b(q,j) are:
1;
1/2,   1/2;
5/12, 8/12, -1/12;
9/24, 19/24, -5/24, 1/24;
... etc.
The denominators are A232853(n).
The numerators are this sequence.
First column's numerators: A002657(n).
Main diagonal's numerators: (-1)^(n+1)*A141417(n).
Row sums are: 1,2,12,24,... (A091137).

			Triangle starts:
1;
1,     1;
5,     8,   -1;
9,    19,   -5,   1;
251, 646, -264, 106, -19;
...
Numerators of
b(0,0)=1, b(1,0)=-(1/2-1)=1/2, b(1,1)=1/2, b(2,0)=(1/3-3/2+2)/2=5/12, b(2,1)=-(1/3-1)=2/3=8/12, b(2,2)=(1/3-1/2)/2=-1/12.

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, (now DGA Maitrise de l'Information 35174 Bruz), 1969, see page 45.

Recurrence:
b(q,j) = (-1)^(q-j)*C(q,j)*b(q,q)+b(q-1,j).
C(q,j)=q!/((q-j)!*j!).

cp's OEIS Frontend

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

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