A165313 Triangle T(n,k) = A091137(k-1) read by rows.

1, 1, 2, 1, 2, 12, 1, 2, 12, 24, 1, 2, 12, 24, 720, 1, 2, 12, 24, 720, 1440, 1, 2, 12, 24, 720, 1440, 60480, 1, 2, 12, 24, 720, 1440, 60480, 120960, 1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 7257600, 1, 2, 12

Offset: 1

Author

Paul Curtz, Sep 14 2009

Keywords

• nonn
• tabl

Comments

From a study of modified initialization formulas in Adams-Bashforth (1855-1883) multisteps method for numerical integration. On p.36, a(i,j) comes from (j!)*a(i,j) = Integral_{u=i,..,i+1} u*(u-1)*...*(u-j+1) du; see p.32.

Then, with i vertical, j horizontal, with unreduced fractions, partial array is:

0) 1, 1/2, -1/12, 1/24, -19/720, 27/1440, ... = 1/log(2)

1) 1, 3/2, 5/12, -1/24, 11/720, -11/1440, ... = 2/log(2)

2) 1, 5/2, 23/12, 9/24, -19/720, 11/1440, ... = 4/log(2)

3) 1, 7/2, 53/12, 55/24, 251/720, -27/1440, ... = 8/log(2)

4) 1, 9/2, 95/12, 161/24, 1901/720, 475/1440, ... = 16/log(2)

5) 1, 11/2, 149/12, 351/24, 6731/720, 4277/1440, ... = 32/log(2)

... [improved by Paul Curtz, Jul 13 2019]

First line: the reduced terms are A002206/A002207, logarithmic or Gregory numbers G(n). The difference between the second line and the first one is 0 together A002206/A002207. This is valid for the next lines. - Paul Curtz, Jul 13 2019

See A141417, A140825, A157982, horizontal numerators: A141047, vertical numerators: A000012, A005408, A140811, A141530, A157411. On p.56, coefficients are s(i,q) = (1/q!)* Integral_{u=-i-1,..,1} u*(u+1)*...*(u+q-1) du.

Unreduced fractions array is:

-1) 1, 1/2, 5/12, 9/24, 251/720, 475/1440, ... = A002657/A091137

0) 2, 0/2, 4/12, 8/24, 232/720, 448/1440, ... = A195287/A091137

1) 3, -3/2, 9/12, 9/24, 243/720, 459/1440, ...

2) 4, -8/2, 32/12, 0/24, 224/720, 448/1440, ...

3) 5, -15/2, 85/12, -55/24, 475/720, 475/1440, ...

...

(on p.56 up to 6)). See A147998. Vertical numerators: A000027, A147998, A152064, A157371, A165281.

From Paul Curtz, Jul 14 2019: (Start)

Difference table from the second line and the first one difference:

1, -1/2, -1/12, -1/24, -19/720, -27/1440, ...

-3/2, 5/12, 1/24, 11/720, 11/1440, ...

23/12, -9/24, -19/720, -11/1440, ...

-55/24, 251/720, 27/1440, ...

1901/720, -475/1440,

-4277/1440, ...

...

Compare the lines to those of the first array.

The verticals are the signed diagonals of the first array. (End)

Examples

			1;
1,2;
1,2,12;
1,2,12,24;
1,2,12,24,720;
		

References

• P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil, 1969.

Crossrefs

Cf. A090624, A091137.

Cf. A000012, A000079, A002657, A005408, A007525, A131920, A140811, A140825, A141047, A141417, A141530, A157411, A157982, A195287.

Cf. A002206, A002207.

Programs

Mathematica

(* a = A091137 *) a[n_] := a[n] = Product[d, {d, Select[Divisors[n]+1, PrimeQ]}]*a[n-1]; a[0]=1; Table[Table[a[k-1], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Dec 18 2014 *)