

A324124


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


6



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
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OFFSET

0,3


LINKS

Table of n, a(n) for n=0..44.
Petros Hadjicostas, Maple program.
Yudell L. Luke, Mechanical quadrature near a singularity, Math. Comp. 6 (1952), 215219.


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 nth row satisfies the equations Sum_{r = 0..n} T(n,nr) * 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, nr)/S(n) = T^*(n, nr)/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

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.
Sequence in context: A198577 A011135 A324775 * A019816 A214562 A280757
Adjacent sequences: A324121 A324122 A324123 * A324125 A324126 A324127


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



