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

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

Author

N. J. A. Sloane, Feb 15 2019

Keywords

• sign
• tabl

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

• Petros Hadjicostas, Maple program.
• Yudell L. Luke, Mechanical quadrature near a singularity, Math. Comp. 6 (1952), 215-219.

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