%I #24 Jan 05 2019 05:20:20
%S 0,1,1,3,10,3,4,11,11,4,11,32,26,32,11,15,43,37,37,43,15,41,118,100,
%T 106,100,118,41,56,161,137,143,143,137,161,56,153,440,374,392,386,392,
%U 374,440,153,209,601,511,535,529,529,535,511,601,209
%N 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.
%C The n-th row common denominator is factorized out and is given by A321119(n).
%C 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.
%D 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.
%H Franck Maminirina Ramaharo, <a href="/A321118/b321118.txt">Rows n = 0..150 of triangle, flattened</a>
%H 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), p. 9-74.
%H John C. Holladay, <a href="https://doi.org/10.1090/S0025-5718-1957-0093894-6">A smoothest curve approximation</a>, Math. Comp. Vol. 11 (1957), 233-243.
%H Peter Köhler, <a href="https://doi.org/10.1007/BF02575942">On the weights of Sard's quadrature formulas</a>, CALCOLO Vol. 25 (1988), 169-186.
%H Leroy F. Meyers and Arthur Sard, <a href="https://doi.org/10.1002/sapm1950291118">Best approximate integration formulas</a>, J. Math. Phys. Vol. 29 (1950), 118-123.
%H Arthur Sard, <a href="https://doi.org/10.2307/2372095">Best approximate integration formulas; best approximation formulas</a>, American Journal of Mathematics Vol. 71 (1949), 80-91.
%H Isaac J. Schoenberg, <a href="https://projecteuclid.org/euclid.bams/1183525790">Spline interpolation and best quadrature formulae</a>, Bull. Amer. Math. Soc. Vol. 70 (1964), 143-148.
%H Frans Schurer, <a href="https://research.tue.nl/en/publications/on-natural-cubic-splines-with-an-application-to-numerical-integra">On natural cubic splines, with an application to numerical integration formulae</a>, EUT report. WSK, Dept. of Mathematics and Computing Science Vol. 70-WSK-04 (1970), 1-32.
%F 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.
%F T(n,k) = T(n,n-k).
%F T(n,k) = 4*T(n-2,k) - T(n-4,k), n >= k + 4.
%F 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.
%F 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.
%F Sum_{k=0..n} T(n,k)/A321119(n) = n.
%e Triangle begins (denominator is factored out):
%e 0; 1/4
%e 1, 1; 1/2
%e 3, 10, 3; 1/8
%e 4, 11, 11, 4; 1/10
%e 11, 32, 26, 32, 11; 1/28
%e 15, 43, 37, 37, 43, 15; 1/38
%e 41, 118, 100, 106, 100, 118, 41; 1/104
%e 56, 161, 137, 143, 143, 137, 161, 56; 1/142
%e 153, 440, 374, 392, 386, 392, 374, 440, 153; 1/388
%e 209, 601, 511, 535, 529, 529, 535, 511, 601, 209; 1/530
%e ...
%e 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)).
%t alpha = (Sqrt[2] + Sqrt[6])/2; T[0,0] = 0;
%t 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)))];
%t a321119[n_] := 2^(-Floor[(n - 1)/2])*((1 - Sqrt[3])^n + (1 + Sqrt[3])^n);
%t Table[FullSimplify[a321119[n]*T[n, k]],{n, 0, 10}, {k, 0, n}] // Flatten
%o (Maxima)
%o (b[0] : 0, b[1] : 1, b[2] : 1, b[3] : 3, b[n] := 4*b[n-2] - b[n-4])$ /* A002530 */
%o d(n) := 2^(-floor((n - 1)/2))*((1 - sqrt(3))^n + (1 + sqrt(3))^n) $ /* A321119 */
%o 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$
%o create_list(ratsimp(T(n, k)), n, 0, 10, k, 0, n);
%Y Cf. A093735, A093736, A100641, A002176, A100640, A321119, A321120, A321121, A321122.
%K nonn,easy,tabl,frac
%O 0,4
%A _Franck Maminirina Ramaharo_, Nov 01 2018