cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

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Author

Paul Curtz, Sep 14 2009

Keywords

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 *)

A195287 a(n) = (A091137(n)/n!) * Integral_{u=-1..1} u*(u+1)*...*(u+n-1) du.

Original entry on oeis.org

2, 0, 4, 8, 232, 448, 18224, 35424, 1036064, 2025472, 130960832, 257072000, 689908475264, 1358275350528, 8031885897472, 15847920983552, 7981032500085248, 15774370258485248, 12448755354530366464
Offset: 0

Views

Author

Paul Curtz, Sep 20 2011

Keywords

Comments

Numerators of the second row of an array based on Adams numerical integration. Take q!*s(m,q) = Integral_{-m-1..1} u*(u+1)*...*(u+q-1) du. a(n) is in the second row (case m=0) numerators of s(m,q) in the comments.
The unreduced array s(m,q), (m=-1,0,1,..., columns q=0,1,2,...) is
1, 1/2, 5/12, 9/24, 251/720, 475/1440, = A002657(n)/A091137(n),
2, 0, 4/12, 8/24, 232/720, 448/1440, = a(n)/A091137(n),
3, -3/2, 9/12, 9/24, 243/720, 459/1440,
4, -8/2, 32/12, 0, 224/720, 448/1440,
5, -15/2, 85/12, -55/24, 475/720, 475/1440,
6, -24/2, 180/12, -216/24, 2376/720, 0.
Column numerators: A000027, -A067998(n), A152064(n), A157371(n), A165281(n).
Page 56 of the reference.
(*) 2/2 = 1,
2/2 + 0 = 1,
2/3 + 0 + 1/3 = 1,
2/4 + 0 + 1/6 + 1/3 = 1. Reduced.

References

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

Programs

  • Maple
    A195287 := proc(n)
        mul(u+i,i=0..n-1) ;
        int(%,u=-1..1) ;
        %/n!*A091137(n) ;
end proc:
seq(A195287(n),n=0..20) ; # R. J. Mathar, Oct 02 2011
  • Mathematica
    (* a7 = A091137 *) a7[n_] := a7[n] = Product[d, {d, Select[Divisors[n] + 1, PrimeQ]}]*a7[n-1]; a7[0]=1; a[n_] := a7[n]/n!*Integrate[ Pochhammer[u, n], {u, -1, 1}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Aug 13 2012 *)

Formula

b(n) = a(n)/A091137(n).
b(0)/2 = 1,
b(0)/2 + b(1) = 1,
b(0)/3 + b(1)/2 + b(2) = 1,
b(0)/4 + b(1)/3 + b(2)/2 + b(3) = 1.
First vertical denominators: A028310(n) + 1. See A104661.
Values in (*).
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