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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. 4
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. 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 2.5.2. 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), 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. FORMULA 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. EXAMPLE 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)). MATHEMATICA 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 PROG (Maxima) (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); CROSSREFS Cf. A093735, A093736, A100641, A002176, A100640, A321119, A321120, A321121, A321122. Sequence in context: A111272 A124692 A091043 * A167790 A010708 A072988 Adjacent sequences:  A321115 A321116 A321117 * A321119 A321120 A321121 KEYWORD nonn,easy,tabl,frac AUTHOR Franck Maminirina Ramaharo, Nov 01 2018 STATUS approved

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Last modified February 23 12:03 EST 2019. Contains 320431 sequences. (Running on oeis4.)