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A165313
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Triangle T(n,k) = A091137(k-1) read by rows.
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4
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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
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OFFSET
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1,3
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COMMENTS
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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)
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) 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, ...
...
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)
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REFERENCES
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P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil, 1969.
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LINKS
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EXAMPLE
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1;
1,2;
1,2,12;
1,2,12,24;
1,2,12,24,720;
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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Cf. A000012, A000079, A002657, A005408, A007525, A131920, A140811, A140825, A141047, A141417, A141530, A157411, A157982, A195287.
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KEYWORD
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AUTHOR
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STATUS
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approved
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