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.

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

Author

Franck Maminirina Ramaharo, Nov 16 2018

Keywords

• nonn
• easy
• tabl
• frac

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.

Examples

			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
  ...
		

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.
• Wikipedia, Newton-Cotes formulas

Crossrefs

Cf. A321122 (Common denominators).

Cf. A093735/A093736 (Newton-Cotes formulas), A100640/A100641 (Cotesian numbers), A321118/A321119 (Holladay-Sard best quadrature formulas).

Programs

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

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));

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.