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A101270
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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.
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3
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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
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OFFSET
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0,6
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COMMENTS
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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
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LINKS
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EXAMPLE
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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;
...
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MAPLE
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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:
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MATHEMATICA
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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]];
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CROSSREFS
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Cf. A324123 (same without the zeros).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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