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Triangle T(n,k) = A091137(k-1) read by rows.
4

%I #48 Sep 07 2019 05:58:08

%S 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,

%T 720,1440,60480,1,2,12,24,720,1440,60480,120960,1,2,12,24,720,1440,

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

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

%C 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.

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

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

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

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

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

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

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

%C ... [improved by _Paul Curtz_, Jul 13 2019]

%C 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

%C 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.

%C Unreduced fractions array is:

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

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

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

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

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

%C ...

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

%C From _Paul Curtz_, Jul 14 2019: (Start)

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

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

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

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

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

%C 1901/720, -475/1440,

%C -4277/1440, ...

%C ...

%C Compare the lines to those of the first array.

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

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

%e 1;

%e 1,2;

%e 1,2,12;

%e 1,2,12,24;

%e 1,2,12,24,720;

%t (* 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 *)

%Y Cf. A090624, A091137.

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

%Y Cf. A002206, A002207.

%K nonn,tabl

%O 1,3

%A _Paul Curtz_, Sep 14 2009