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A321121
Triangle read by rows: T(n,k) is the unreduced numerator of the k-th weight in the quadrature rule for parabolic runout spline with respect to a mesh of n + 1 points.
3
0, 1, 1, 1, 4, 1, 3, 9, 9, 3, 13, 44, 30, 44, 13, 35, 115, 90, 90, 115, 35, 16, 53, 40, 46, 40, 53, 16, 131, 433, 330, 366, 366, 330, 433, 131, 179, 592, 450, 504, 486, 504, 450, 592, 179, 163, 539, 410, 458, 446, 446, 458, 410, 539, 163, 668, 2209, 1680, 1878, 1824, 1842, 1824, 1878, 1680, 2209, 668
OFFSET
0,5
COMMENTS
The weights in this quadrature rule are T(n,k)/A321122(n), 0 <= k <= n. For n = 1, 2, 3, we obtain the trapezoid rule, Simpson's rule, and Simpson's 3/8 rule, respectively.
REFERENCES
Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967. See p. 47, Table 2.5.3.
LINKS
Franck Maminirina Ramaharo, Rows n = 0..150 of triangle, flattened
Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, Chapter II. The Cubic Spline, Mathematics in Science and Engineering Volume 38 (1967), pp. 9-74.
FORMULA
T(n,k) = T(n,n-k).
T(0,0) = 0 and T(n,k) = A093735(n,k) for n = 1, 2, 3.
Let s = -2 + sqrt(3), and define e(n) = s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n)), f(n,k) = 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n)), and w(n,0) = 1/4 + e(n)/6, w(n,1) = 2 - (1 + 1/6)*e(n), w(n,k) = 1 + f(n,k)/4 for 2 <= k <= n - 2. Then T(n,k) = A321122(n)*w(n,k) for 0 <= k <= n, n >= 3.
EXAMPLE
Triangle begins (denominator is factored out):
0; 1/4
1, 1; 1/2
1, 4, 1; 1/3
3, 9, 9, 3; 1/8
13, 44, 30, 44, 13; 1/36
35, 115, 90, 90, 115, 35; 1/96
16, 53, 40, 46, 40, 53, 16; 1/44
131, 433, 330, 366, 366, 330, 433, 131; 1/360
179, 592, 450, 504, 486, 504, 450, 592, 179; 1/492
163, 539, 410, 458, 446, 446, 458, 410, 539, 163; 1/448
...
MATHEMATICA
s = -2 + Sqrt[3];
e[n_] := s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n));
f[n_, k_] := 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n));
w[n_, k_] := If[k == 0 || k == n, 1/4 + e[n]/6, If[k == 1 || k == n - 1, 2 - (1 + 1/6)*e[n], 1 + f[n, k]/4]];
a321122[n_] := LCM @@ Table[Denominator[FullSimplify[w[n, k]]], {k, 0, n}]
Join[{0, 1, 1, 1, 4, 1}, Table[FullSimplify[a321122[n]*w[n, k]], {n, 3, 12}, {k, 0, n}]] // Flatten
PROG
(Maxima)
s : -2 + sqrt(3)$
e(n) := s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n))$
f(n, k) := 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n))$
w(n, k) := if k = 0 or k = n then 1/4 + e(n)/6 else if k = 1 or k = n - 1 then 2 - (1 + 1/6)*e(n) else 1 + f(n, k)/4$
a321122(n) := lcm(makelist(denom(fullratsimp(w(n, k))), k, 0, n))$
append([0, 1, 1, 1, 4, 1], create_list(fullratsimp(a321122(n)*w(n, k)), n, 3, 12, k, 0, n));
CROSSREFS
Cf. A321122 (Common denominators).
Cf. A093735/A093736 (Newton-Cotes formulas), A100640/A100641 (Cotesian numbers), A321118/A321119 (Holladay-Sard best quadrature formulas).
Sequence in context: A196770 A154182 A231921 * A093735 A298918 A156224
KEYWORD
nonn,easy,tabl,frac
AUTHOR
STATUS
approved