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Coefficients for numerical integration.
(Formerly M1702 N0671)
+10
7
2, 6, 34, 250, 972, 15498, 766808, 5961306, 54891535, 2488870076
COMMENTS
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).
CROSSREFS
Main diagonal of A324124 (except for a(7)).
Triangle T(n,k), read by rows: coefficients for numerical integration near a singularity (n >= 0 and 0 <= k <= n).
+10
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
FORMULA
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;
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.
1, 3, 15, 105, 945, 4158, 75075, 579150, 34459425, 339489150, 16499172690, 1724913508500, 878428175625, 131358797955000, 252664626678750, 3891675750482520, 221643095476699771875, 208605266331011550000, 32498195069133543750, 807994071847270157850000, 125885476393804690593030
COMMENTS
These are the quantities D_n in Luke (1952). See the table on p. 217 of his paper. We use a(7) = 579150, which is a divisor of D_7 = 4054050. For an explanation of this change, see the comments for array A324124. (This modification does not change the values of the fractions gamma_r(n)/D_n used in Eq. (1) of Luke (1952). It is needed to give a unique definition of the terms of the array A324124.)
1, 2, 6, 34, 250, 972, 15498, 109544, 5961306, 54891535, 2488870076, 246264587430, 118503860254, 16917328320424, 31020551370600, 459016311081816, 25084010473396186126, 22789919291848918932, 3423598216929042597, 82480361346228654485320, 12440086312584102532500, 121292379506812780007192
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.
Coefficients for numerical integration.
(Formerly M4533 N1922)
+10
5
1, 8, 45, 416, 1685, 31032, 1603182, 13856896, 132843888, 6551143600
COMMENTS
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).
CROSSREFS
Secondary diagonal of A324124 (except for a(7)).
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