A093735 -id:A093735 - cp's OEIS frontend

A321118 T(n,k) = A321119(n) - (-1)^kA321119(n-2k)/2 for 0 < k < n, with T(0,0) = 0 and T(n,0) = T(n,n) = A002530(n+1) for n > 0, triangle read by rows; unreduced numerator of the weights of Holladay-Sard's quadrature formula.

0, 1, 1, 3, 10, 3, 4, 11, 11, 4, 11, 32, 26, 32, 11, 15, 43, 37, 37, 43, 15, 41, 118, 100, 106, 100, 118, 41, 56, 161, 137, 143, 143, 137, 161, 56, 153, 440, 374, 392, 386, 392, 374, 440, 153, 209, 601, 511, 535, 529, 529, 535, 511, 601, 209

Offset: 0

Franck Maminirina Ramaharo, Nov 01 2018

The n-th row common denominator is factorized out and is given by A321119(n).

Given a continuous function f over the interval [0,n], the best quadrature formula in the sense of Holladay-Sard is given by Integral_{x=0..n} f(x) dx = Sum_{k=0..n} T(n,k)*f(k)/A321119(n). The formula is exact if f belongs to the class of natural cubic splines.

			Triangle begins (denominator is factored out):
    0;                                                 1/4
    1,   1;                                            1/2
    3,  10,   3;                                       1/8
    4,  11,  11,   4;                                  1/10
   11,  32,  26,  32,  11;                             1/28
   15,  43,  37,  37,  43,  15;                        1/38
   41, 118, 100, 106, 100, 118,  41;                   1/104
   56, 161, 137, 143, 143, 137, 161,  56;              1/142
  153, 440, 374, 392, 386, 392, 374, 440, 153;         1/388
  209, 601, 511, 535, 529, 529, 535, 511, 601, 209;    1/530
  ...
If f is a continuous function over the interval [0,3], then the quadrature formula yields Integral_{x=0..3} f(x) d(x) = (1/10)*(4*f(0) + 11*f(1) + 11*f(2) + 4*f(3)).

Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967. See p. 47, Table 2.5.2.

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), p. 9-74.
John C. Holladay, A smoothest curve approximation, Math. Comp. Vol. 11 (1957), 233-243.
Peter Köhler, On the weights of Sard's quadrature formulas, CALCOLO Vol. 25 (1988), 169-186.
Leroy F. Meyers and Arthur Sard, Best approximate integration formulas, J. Math. Phys. Vol. 29 (1950), 118-123.
Arthur Sard, Best approximate integration formulas; best approximation formulas, American Journal of Mathematics Vol. 71 (1949), 80-91.
Isaac J. Schoenberg, Spline interpolation and best quadrature formulae, Bull. Amer. Math. Soc. Vol. 70 (1964), 143-148.
Frans Schurer, On natural cubic splines, with an application to numerical integration formulae, EUT report. WSK, Dept. of Mathematics and Computing Science Vol. 70-WSK-04 (1970), 1-32.

Cf. A093735, A093736, A100641, A002176, A100640, A321119, A321120, A321121, A321122.

alpha = (Sqrt[2] + Sqrt[6])/2; T[0,0] = 0;
T[n_, k_] := If[n > 0 && k == 0 || k == n, (alpha^(n + 1) - (-alpha)^(-(n + 1)))/(2*Sqrt[6]*(alpha^n + (-alpha)^(-n))), 1 - (-1)^k*(alpha^(n - 2*k) + (-alpha)^(2*k - n))/(2*(alpha^n + (-alpha)^(-n)))];
a321119[n_] := 2^(-Floor[(n - 1)/2])*((1 - Sqrt[3])^n + (1 + Sqrt[3])^n);
Table[FullSimplify[a321119[n]*T[n, k]],{n, 0, 10}, {k, 0, n}] // Flatten

(b[0] : 0, b[1] : 1, b[2] : 1, b[3] : 3, b[n] := 4*b[n-2] - b[n-4])$ /* A002530 */
d(n) := 2^(-floor((n - 1)/2))*((1 - sqrt(3))^n + (1 + sqrt(3))^n) $ /* A321119 */
T(n, k) := if n = 0 and k = 0 then 0 else if n > 0 and k = 0 or k = n then b[n + 1] else d(n) - (-1)^k*d(n - 2*k)/2$
create_list(ratsimp(T(n, k)), n, 0, 10, k, 0, n);

T(n,k)/A321119(n) = (alpha^(n + 1) - (-alpha)^(-(n + 1)))/(2*sqrt(6)*(alpha^n + (-alpha)^(-n))) if k = 0 or k = n, and 1 - (-1)^k*(alpha^(n - 2*k) + (-alpha)^(2*k - n))/(2*(alpha^n + (-alpha)^(-n))) if 0 < k < n, where alpha = (sqrt(2) + sqrt(6))/2.
T(n,k) = T(n,n-k).
T(n,k) = 4*T(n-2,k) - T(n-4,k), n >= k + 4.
T(2*n+2,k)*A001834(n+1) = A001834(n)*T(2*n,k) + 2*A003500(n)*T(2*n+1,k) for k < 2*n.
T(2*n+3,k)*A003500(n+1) = A003500(n)*T(2*n+1,k) + 2*A001834(n+1)*T(2*n+2,k) for k < 2*n + 1.
Sum_{k=0..n} T(n,k)/A321119(n) = 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.

Original entry on oeis.org

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

Franck Maminirina Ramaharo, Nov 16 2018

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.

			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
  ...

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.

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

Cf. A321122 (Common denominators).

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

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

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));

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.

cp's OEIS Frontend

A093736 Denominators of Newton-Cotes formulas.

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A321118 T(n,k) = A321119(n) - (-1)^kA321119(n-2k)/2 for 0 < k < n, with T(0,0) = 0 and T(n,0) = T(n,n) = A002530(n+1) for n > 0, triangle read by rows; unreduced numerator of the weights of Holladay-Sard's quadrature formula.

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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.

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cp's OEIS Frontend

A093736 Denominators of Newton-Cotes formulas.

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A321118 T(n,k) = A321119(n) - (-1)^k*A321119(n-2*k)/2 for 0 < k < n, with T(0,0) = 0 and T(n,0) = T(n,n) = A002530(n+1) for n > 0, triangle read by rows; unreduced numerator of the weights of Holladay-Sard's quadrature formula.

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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.

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A321118 T(n,k) = A321119(n) - (-1)^kA321119(n-2k)/2 for 0 < k < n, with T(0,0) = 0 and T(n,0) = T(n,n) = A002530(n+1) for n > 0, triangle read by rows; unreduced numerator of the weights of Holladay-Sard's quadrature formula.