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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.

%I A321121 #23 Jan 05 2019 05:20:41
%S A321121 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,
%T A321121 40,53,16,131,433,330,366,366,330,433,131,179,592,450,504,486,504,450,
%U A321121 592,179,163,539,410,458,446,446,458,410,539,163,668,2209,1680,1878,1824,1842,1824,1878,1680,2209,668
%N 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.
%C A321121 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.
%D A321121 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.
%H A321121 Franck Maminirina Ramaharo, <a href="/A321121/b321121.txt">Rows n = 0..150 of triangle, flattened</a>
%H A321121 Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, <a href="https://doi.org/10.1016/S0076-5392(08)61988-8">Chapter II. The Cubic Spline</a>, Mathematics in Science and Engineering Volume 38 (1967), pp. 9-74.
%H A321121 Wikipedia, <a href="https://en.wikipedia.org/wiki/Newton%E2%80%93Cotes_formulas">Newton-Cotes formulas</a>
%F A321121 T(n,k) = T(n,n-k).
%F A321121 T(0,0) = 0 and T(n,k) = A093735(n,k) for n = 1, 2, 3.
%F A321121 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.
%e A321121 Triangle begins (denominator is factored out):
%e A321121     0;                                                 1/4
%e A321121     1,   1;                                            1/2
%e A321121     1,   4,   1;                                       1/3
%e A321121     3,   9,   9,   3;                                  1/8
%e A321121    13,  44,  30,  44,  13;                             1/36
%e A321121    35, 115,  90,  90, 115,  35;                        1/96
%e A321121    16,  53,  40,  46,  40,  53,  16;                   1/44
%e A321121   131, 433, 330, 366, 366, 330, 433, 131;              1/360
%e A321121   179, 592, 450, 504, 486, 504, 450, 592, 179;         1/492
%e A321121   163, 539, 410, 458, 446, 446, 458, 410, 539, 163;    1/448
%e A321121   ...
%t A321121 s = -2 + Sqrt[3];
%t A321121 e[n_] := s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n));
%t A321121 f[n_, k_] := 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n));
%t A321121 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]];
%t A321121 a321122[n_] := LCM @@ Table[Denominator[FullSimplify[w[n, k]]], {k, 0, n}]
%t A321121 Join[{0, 1, 1, 1, 4, 1}, Table[FullSimplify[a321122[n]*w[n, k]], {n, 3, 12}, {k, 0, n}]] // Flatten
%o A321121 (Maxima)
%o A321121 s : -2 + sqrt(3)$
%o A321121 e(n) := s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n))$
%o A321121 f(n, k) := 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n))$
%o A321121 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$
%o A321121 a321122(n) := lcm(makelist(denom(fullratsimp(w(n, k))), k, 0, n))$
%o A321121 append([0, 1, 1, 1, 4, 1], create_list(fullratsimp(a321122(n)*w(n, k)), n, 3, 12, k, 0, n));
%Y A321121 Cf. A321122 (Common denominators).
%Y A321121 Cf. A093735/A093736 (Newton-Cotes formulas), A100640/A100641 (Cotesian numbers), A321118/A321119 (Holladay-Sard best quadrature formulas).
%K A321121 nonn,easy,tabl,frac
%O A321121 0,5
%A A321121 _Franck Maminirina Ramaharo_, Nov 16 2018