OFFSET
0,3
LINKS
Petros Hadjicostas, Maple program.
Yudell L. Luke, Mechanical quadrature near a singularity, Math. Comp. 6 (1952), 215-219.
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)
EXAMPLE
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)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
N. J. A. Sloane, Feb 15 2019
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
STATUS
approved