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.

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

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

Table of n, a(n) for n=0..65.

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

Adjacent sequences: A101267 A101268 A101269 * A101271 A101272 A101273

Keyword

sign,tabl

Author

Emeric Deutsch, Jan 24 2005

Extensions

T(0,0) = 1 prepended by Petros Hadjicostas, Oct 28 2019

Status

approved