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A101270 T(n, k) is the coefficient of z^k in the numerator of the polynomial part of z^n*exp(-n*s), where s = hypergeom([1, 1, 3/2], [2, 5/2], 1/z^2)/(6z^2); related to Chebyshev's quadrature. Triangle read by rows, T(n,k) for 0 <= k <= n. 3
1, 0, 1, -1, 0, 3, 0, -1, 0, 2, 1, 0, -30, 0, 45, 0, 7, 0, -60, 0, 72, -1, 0, 21, 0, -105, 0, 105, 0, -149, 0, 2142, 0, -7560, 0, 6480, -43, 0, -2220, 0, 20790, 0, -56700, 0, 42525, 0, 53, 0, -2280, 0, 15120, 0, -33600, 0, 22400, -43, 0, 561, 0, -9900, 0, 49896, 0, -93555, 0, 56133 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
Without the zeros and with the powers of the coefficients in reverse order (in each row), this array is essentially the same as A324123. For Maple programs to generate the rows of this array, see the link and the program section. - Petros Hadjicostas, Oct 28 2019
LINKS
Petros Hadjicostas, Alternative Maple program.
H. E. Salzer, Tables for facilitating the use of Chebyshev's quadrature formula, Journal of Mathematics and Physics, 26 (1947), 191-194.
Eric Weisstein's World of Mathematics, Chebyshev Quadrature.
EXAMPLE
T(4,0) = 1, T(4,1)=0, T(4,2) = -30, T(4,3) = 0, T(4,4) = 45 because
z^4*exp(-4s) = z^4 - 2*z^2/3 + 1/45 - 32/(2835*z^2) + O(1/z^4) = (45*z^4 - 30*z^2 + 1)/45 - 32/(2835*z^2) + O(1/z^4).
Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
1
0, 1;
-1, 0, 3;
0, -1, 0, 2;
1, 0, -30, 0, 45;
0, 7, 0, -60, 0, 72;
-1, 0, 21, 0, -105, 0, 105;
...
MAPLE
gf := n -> exp(n*(arctanh(z)/z + 1/2*log(1-z^2) - 1)):
ser := n -> series(gf(n), z, n + 2):
g := n -> ilcm(seq(denom(coeff(ser(n), z, k)), k = 0..n)):
a := proc(n) local S; S:=ser(n); seq(g(n)*coeff(S, z, n-m), m=0..n) end:
seq(a(n), n = 0..10); # Petros Hadjicostas, Oct 28 2019
MATHEMATICA
row[0] = {1}; row[1] = {0, 1}; row[n_] := row[n] = Select[ Normal[z^n*Exp[-n*HypergeometricPFQ[{1, 1, 3/2}, {2, 5/2}, 1/z^2]/(6 z^2)] + O[z, Infinity]^n], PolynomialQ[#, z]&] // Together // Numerator // CoefficientList[#, z]&;
T[n_, k_] := row[n][[k + 1]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 26 2023 *)
CROSSREFS
T(n, n) = A002680(n).
Cf. A324123 (same without the zeros).
Sequence in context: A356679 A356681 A093684 * A351097 A155522 A007524
KEYWORD
sign,tabl
AUTHOR
Emeric Deutsch, Jan 24 2005
EXTENSIONS
T(0,0) = 1 prepended by Petros Hadjicostas, Oct 28 2019
STATUS
approved

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Last modified March 19 06:47 EDT 2024. Contains 370953 sequences. (Running on oeis4.)