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

%I A165313 #48 Sep 07 2019 05:58:08
%S A165313 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 A165313 720,1440,60480,1,2,12,24,720,1440,60480,120960,1,2,12,24,720,1440,
%U A165313 60480,120960,3628800,1,2,12,24,720,1440,60480,120960,3628800,7257600,1,2,12
%N A165313 Triangle T(n,k) = A091137(k-1) read by rows.
%C A165313 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 A165313 Then, with i vertical, j horizontal, with unreduced fractions, partial array is:
%C A165313 0) 1,  1/2,  -1/12,   1/24,  -19/720,   27/1440, ... =  1/log(2)
%C A165313 1) 1,  3/2,   5/12,  -1/24,   11/720,  -11/1440, ... =  2/log(2)
%C A165313 2) 1,  5/2,  23/12,   9/24,  -19/720,   11/1440, ... =  4/log(2)
%C A165313 3) 1,  7/2,  53/12,  55/24,  251/720,  -27/1440, ... =  8/log(2)
%C A165313 4) 1,  9/2,  95/12, 161/24, 1901/720,  475/1440, ... = 16/log(2)
%C A165313 5) 1, 11/2, 149/12, 351/24, 6731/720, 4277/1440, ... = 32/log(2)
%C A165313 ... [improved by _Paul Curtz_, Jul 13 2019]
%C A165313 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 A165313 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 A165313 Unreduced fractions array is:
%C A165313 -1) 1,   1/2,  5/12,   9/24, 251/720, 475/1440, ... = A002657/A091137
%C A165313 0)  2,   0/2,  4/12,   8/24, 232/720, 448/1440, ... = A195287/A091137
%C A165313 1)  3,  -3/2,  9/12,   9/24, 243/720, 459/1440, ...
%C A165313 2)  4,  -8/2, 32/12,   0/24, 224/720, 448/1440, ...
%C A165313 3)  5, -15/2, 85/12, -55/24, 475/720, 475/1440, ...
%C A165313 ...
%C A165313 (on p.56 up to 6)). See A147998. Vertical numerators: A000027, A147998, A152064, A157371, A165281.
%C A165313 From _Paul Curtz_, Jul 14 2019: (Start)
%C A165313 Difference table from the second line and the first one difference:
%C A165313 1,                -1/2,      -1/12,    -1/24,    -19/720, -27/1440, ...
%C A165313 -3/2,             5/12,       1/24,    11/720,    11/1440, ...
%C A165313 23/12,           -9/24,    -19/720,  -11/1440, ...
%C A165313 -55/24,        251/720,    27/1440, ...
%C A165313 1901/720,    -475/1440,
%C A165313 -4277/1440, ...
%C A165313 ...
%C A165313 Compare the lines to those of the first array.
%C A165313 The verticals are the signed diagonals of the first array. (End)
%D A165313 P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil, 1969.
%e A165313 1;
%e A165313 1,2;
%e A165313 1,2,12;
%e A165313 1,2,12,24;
%e A165313 1,2,12,24,720;
%t A165313 (* 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 A165313 Cf. A090624, A091137.
%Y A165313 Cf. A000012, A000079, A002657, A005408, A007525, A131920, A140811, A140825, A141047, A141417, A141530, A157411, A157982, A195287.
%Y A165313 Cf. A002206, A002207.
%K A165313 nonn,tabl
%O A165313 1,3
%A A165313 _Paul Curtz_, Sep 14 2009