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.

Showing 1-3 of 3 results.

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

Views

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 *)

A165281 a(n) = (n+1)*(6*n^4 - 51*n^3 + 161*n^2 - 251*n + 251).

Original entry on oeis.org

251, 232, 243, 224, 475, 2376, 9107, 26368, 63099, 132200, 251251, 443232, 737243, 1169224, 1782675, 2629376, 3770107, 5275368, 7226099, 9714400, 12844251, 16732232, 21508243, 27316224, 34314875, 42678376, 52597107, 64278368, 77947099
Offset: 0

Views

Author

Paul Curtz, Sep 13 2009

Keywords

Comments

The sequence is the numerators of the fifth column of the array on page 56 of the reference. The denominators are A091137(4)=720.
The sequence is the binomial transform of the quasi-finite 251, -19, 30, -60, 360, 720, 0, 0, 0, 0, ...
The fifth differences are (constant) 720; the fourth differences are 720*n + 360.

References

  • P. Curtz, Integration numerique des systemes differentiels a conditions initiales, C.C.S.A., Arcueil, 1969.

Links

Crossrefs

Cf. A157371, A152064.

Programs

  • Magma
    [(n+1)*(6*n^4-51*n^3+161*n^2-251*n+251): n in [0..30]]; // Vincenzo Librandi, Aug 07 2011
  • Mathematica
    Table[(n+1)(6n^4-51n^3+161n^2-251n+251),{n,0,30}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{251,232,243,224,475,2376},30] (* Harvey P. Dale, Aug 20 2014 *)

Formula

a(n) mod 10 = A010879(n+1).
a(n+1) - a(n) = A157411(n).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: ( 251 - 1274*x + 2616*x^2 - 2774*x^3 + 1901*x^4 ) / (x-1)^6. - R. J. Mathar, Jul 06 2011

A157982 Triangle T(n,m) read by rows which contains the coefficients [x^m] of the polynomial generating the numerators of the column A140825(.,n).

Original entry on oeis.org

1, 1, 2, -1, 0, 6, 1, 0, -6, 4, -19, 0, 120, -120, 30, 27
Offset: 0

Views

Author

Paul Curtz, Mar 10 2009

Keywords

Comments

The first five polynomials describing the first five columns of A140825 are in A000012, A005408, A140811, A141530 and A157411.

Examples

			1;
1,2;     # 2n+1
-1,0,6;  # 6n^2-1
1,0,-6,4;  # 4n^3-6n^2+1, A141530
-19,0,120,-120,30;  # 30n^4-120n^3+120n^2-19, A157411

References

  • P Curtz Integration numerique des systemes differentiels a conditions initiales, C.C.S.A., Arcueil, 1969, p.36.

Crossrefs

Cf. A141417.
Showing 1-3 of 3 results.

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