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

%I A324124 #52 Oct 30 2019 03:45:09
%S A324124 1,1,2,1,8,6,8,18,45,34,31,224,24,416,250,161,460,840,40,1685,972,
%T A324124 1588,12312,-3870,26480,-7965,31032,15498,14445,49784,79086,-41160,
%U A324124 214865,-76440,229026,109544,530095,4469632,-3257376,14249344,-13403240,20311680,-8258912,13856896,5961306
%N A324124 Triangle T(n,k), read by rows: coefficients for numerical integration near a singularity (n >= 0 and 0 <= k <= n).
%H A324124 Petros Hadjicostas, <a href="/A324124/a324124.txt">Maple program</a>.
%H A324124 Yudell L. Luke, <a href="https://doi.org/10.1090/S0025-5718-1952-0050993-8">Mechanical quadrature near a singularity</a>, Math. Comp. 6 (1952), 215-219.
%F A324124 From _Petros Hadjicostas_, Oct 29 2019: (Start)
%F A324124 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.
%F A324124 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!]
%F A324124 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.
%F A324124 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).
%F A324124 (End)
%e A324124 Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
%e A324124       1;
%e A324124       1,     2;
%e A324124       1,     8,     6;
%e A324124       8,    18,    45,     34;
%e A324124      31,   224,    24,    416,    250;
%e A324124     161,   460,   840,     40,   1685,    972;
%e A324124    1588, 12312, -3870,  26480,  -7965,  31032,  15498;
%e A324124   14445, 49784, 79086, -41160, 214865, -76440, 229026, 109544;
%e A324124   ... [Edited by _Petros Hadjicostas_, Oct 29 2019]
%e A324124 From _Petros Hadjicostas_, Oct 29 2019: (Start)
%e A324124 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:
%e A324124 8*3^0 + 18*2^0 + 45*1^0 + 34*0^0 = S(3)*3^0/(2*0+1);
%e A324124 8*3^1 + 18*2^1 + 45*1^1 + 34*0^1 = S(3)*3^1/(2*1+1);
%e A324124 8*3^2 + 18*2^2 + 45*1^2 + 34*0^2 = S(3)*3^2/(2*2+1);
%e A324124 8*3^3 + 18*2^3 + 45*1^3 + 34*0^3 = S(3)*3^3/(2*3+1).
%e A324124 (End)
%Y A324124 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.
%Y A324124 Row sums appear in A328866.
%K A324124 sign,tabl
%O A324124 0,3
%A A324124 _N. J. A. Sloane_, Feb 15 2019
%E A324124 Name edited by and more terms from _Petros Hadjicostas_, Oct 29 2019
%E A324124 Row n=7 from Luke (1952) was modified by _Petros Hadjicostas_, Oct 29 2019