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 1-5 of 5 results.

A002685 Coefficients for numerical integration.

Original entry on oeis.org

2, 6, 34, 250, 972, 15498, 766808, 5961306, 54891535, 2488870076
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

From Petros Hadjicostas, Oct 29 2019: (Start)
This is the main diagonal of A324124 (which is essentially the table on p. 217 in Luke (1952)), except that a(7) = 766808 must be replaced with 766808/7 = 109544. This is necessary to give a unique definition to the terms of A324124. For more information, see the comments for A324124.
Luke (1952) is not wrong (since 766808/4054050 = 109544/579150), but his integers for the case n = 7 have to modified as mentioned in the documentation of array A324124. For an improved version of this sequence, see A328884. (End)

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002686, A328866, A328884, A328885.
Main diagonal of A324124 (except for a(7)).

A324124 Triangle T(n,k), read by rows: coefficients for numerical integration near a singularity (n >= 0 and 0 <= k <= n).

Original entry on oeis.org

1, 1, 2, 1, 8, 6, 8, 18, 45, 34, 31, 224, 24, 416, 250, 161, 460, 840, 40, 1685, 972, 1588, 12312, -3870, 26480, -7965, 31032, 15498, 14445, 49784, 79086, -41160, 214865, -76440, 229026, 109544, 530095, 4469632, -3257376, 14249344, -13403240, 20311680, -8258912, 13856896, 5961306
Offset: 0

Views

Author

N. J. A. Sloane, Feb 15 2019

Keywords

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
      1;
      1,     2;
      1,     8,     6;
      8,    18,    45,     34;
     31,   224,    24,    416,    250;
    161,   460,   840,     40,   1685,    972;
   1588, 12312, -3870,  26480,  -7965,  31032,  15498;
  14445, 49784, 79086, -41160, 214865, -76440, 229026, 109544;
  ... [Edited by _Petros Hadjicostas_, Oct 29 2019]
From _Petros Hadjicostas_, Oct 29 2019: (Start)
Consider row n=3. We have T(n,0) = 8, T(n,1) = 18, T(n,2) = 45, and T(n,3) = 34 with S(n) = 8 + 18 + 45 + 34 = 105 = A328866(3). We then have the following four equations:
8*3^0 + 18*2^0 + 45*1^0 + 34*0^0 = S(3)*3^0/(2*0+1);
8*3^1 + 18*2^1 + 45*1^1 + 34*0^1 = S(3)*3^1/(2*1+1);
8*3^2 + 18*2^2 + 45*1^2 + 34*0^2 = S(3)*3^2/(2*2+1);
8*3^3 + 18*2^3 + 45*1^3 + 34*0^3 = S(3)*3^3/(2*3+1).
(End)

Links

Crossrefs

A002685 and A002686 give the first two diagonals (except for the elements of row n=7 of this array). Improved versions of these two sequences appear in A328884 and A328885, respectively.
Row sums appear in A328866.

Formula

From Petros Hadjicostas, Oct 29 2019: (Start)
Let S(n) = Sum_{k = 0..n} T(n,k) = A328866(n) for n >= 0. Then the n-th row satisfies the equations Sum_{r = 0..n} T(n,n-r) * r^m = S(n)*n^m/(2*m+1) for m = 0, 1, ..., n.
Note that, if c is a positive integer and T^*(n,k) := c * T(n,k) and S^*(n) := Sum_{k = 0..n} T^*(n,k) = c * S(n), then the new array T^*(n,k) satisfies the same equations and can also be used for the quadrature described in Luke (1952). The reason is that T^*(n,k)/S^*(n) = T(n,k)/S(n) and in Eq. (1), on p. 215 of his paper, what matters is the ratio gamma_r^(n)/D_n = T(n, n-r)/S(n) = T^*(n, n-r)/S^*(n). [Note that the only place in Luke (1952) where gamma_r^(n) is not divided by D_n is in Eq. (6) on p. 216, but that is clearly a typo!]
To make the definition of the array T(n,k) unique, we need to impose a restriction on the sum S(n). Since in each row we are dealing with the fractions T(n,k)/S(n) for k = 0..n and Sum_{k = 0..n} T(n,k)/S(n) = 1, a reasonable assumption is to require S(n) to be the LCM of the denominators of the fractions (T(n,k)/S(n), k = 0..n) in lowest terms. This is done by Luke (1952) (on p. 217 of his paper) for 1 <= n <= 10 except (unfortunately) for n = 7.
For n = 7, Luke (1952) uses the fractions (101115, 348488, 553602, -288120, 1504055, -535080, 1603182, 766808)/4054050, which in lowest terms become (107/4290, 24892/289575, 13181/96525, -1372/19305, 42973/115830, -196/1485, 38171/96525, 54772/289575). The LCM of these denominators is 579150, which is a divisor of 4054050. Putting these fractions under the common denominator 579150, we get (14445, 49784, 79086, -41160, 214865, -76440, 229026, 109544)/579150. We use the numerators of these fractions in this array for (T(n=7, k): k = 0..7).
(End)

Extensions

Name edited by and more terms from Petros Hadjicostas, Oct 29 2019
Row n=7 from Luke (1952) was modified by Petros Hadjicostas, Oct 29 2019

A328884 Improved version of A002685; main diagonal of A324124.

Original entry on oeis.org

1, 2, 6, 34, 250, 972, 15498, 109544, 5961306, 54891535, 2488870076, 246264587430, 118503860254, 16917328320424, 31020551370600, 459016311081816, 25084010473396186126, 22789919291848918932, 3423598216929042597, 82480361346228654485320, 12440086312584102532500, 121292379506812780007192
Offset: 0

Views

Author

Petros Hadjicostas, Oct 29 2019

Keywords

Comments

Sequence A002685 has only 10 terms (from n = 1 to n = 10). We have a(n) = A002685(n) for 1 <= n <= 10 except for n = 7: a(7) = 109544 while A002685(7) = 766808 (which is a(7) * 7 = 109544 * 7). For an explanation, see the comments for array A324124. This is necessary to give a unique definition to the terms of array A324124.

Links

Crossrefs

Cf. A002685, A002686, A324124, A328866, A328885.

A328885 Improved version of A002686; secondary main diagonal of A324124.

Original entry on oeis.org

1, 8, 45, 416, 1685, 31032, 229026, 13856896, 132843888, 6551143600, 672567897859, 346984064352, 51233511973865, 99704578224832, 1521740934249300, 87595020192056292864, 81891555365769626429, 12883000343953567644, 318718123756704667891346, 50107182570391966040000, 500792393469899639083500
Offset: 1

Views

Author

Petros Hadjicostas, Oct 29 2019

Keywords

Comments

Sequence A002686 has only 10 terms (from n = 1 to n = 10). We have a(n) = A002686(n) for 1 <= n <= 10 except for n = 7: a(7) = 229026 while A002686(7) = 1603182 (which is a(7) * 7 = 229026 * 7). For an explanation, see the comments for array A324124. This is necessary to give a unique definition to the terms of array A324124.

Links

Crossrefs

Cf. A002685, A002686, A324124, A328866, A328884.

A002686 Coefficients for numerical integration.

Original entry on oeis.org

1, 8, 45, 416, 1685, 31032, 1603182, 13856896, 132843888, 6551143600
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

From Petros Hadjicostas, Oct 29 2019: (Start)
This is the secondary main diagonal of A324124 (which is essentially Table I on p. 217 in Luke (1952)), except that a(7) = 1603182 must be replaced with 1603182/7 = 229026. This is necessary to give a unique definition to the terms of A324124. For more information, see the comments for A324124.
Luke (1952) is not wrong (since 1603182/4054050 = 229026/579150), but his integers for the case n = 7 have to modified as mentioned in the documentation of array A324124. For an improved version of this sequence, see A328885. (End)

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002685, A328866, A328884, A328885.
Secondary diagonal of A324124 (except for a(7)).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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