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A344909 T(n, k) = numerator([x^k] [z^n] ((1 - i*z)/(1 + i*z))^(i*x)*(1 + z^2)^(-3/4)). Numerators of the coefficients of the symmetric Meixner-Pollaczek polynomials P^(3/4)_{n}(x, Pi/2). Triangle read by rows, T(n, k) for 0 <= k <= n.

%I A344909 #22 Apr 05 2022 05:24:37
%S A344909 1,0,2,-3,0,2,0,-13,0,4,21,0,-17,0,2,0,177,0,-7,0,4,-77,0,2401,0,-25,
%T A344909 0,4,0,-4987,0,1123,0,-29,0,8,1155,0,-24749,0,1499,0,-11,0,2,0,718657,
%U A344909 0,-341521,0,1157,0,-74,0,4,-4389,0,6361429,0,-495469,0,2411,0,-41,0,4
%N A344909 T(n, k) = numerator([x^k] [z^n] ((1 - i*z)/(1 + i*z))^(i*x)*(1 + z^2)^(-3/4)). Numerators of the coefficients of the symmetric Meixner-Pollaczek polynomials P^(3/4)_{n}(x, Pi/2). Triangle read by rows, T(n, k) for 0 <= k <= n.
%H A344909 R. Koekoek, P. A. Lesky, and R. F. Swarttouw, <a href="https://doi.org/10.1007/978-3-642-05014-5">Hypergeometric Orthogonal Polynomials and Their q-Analogues.</a> Springer, 2010. (p. 213-216.)
%H A344909 Alexey Kuznetsov, <a href="https://doi.org/10.4153/CMB-2008-055-0">Expansion of the Riemann Xi Function in Meixner-Pollaczek Polynomials</a>. Canadian Mathematical Bulletin, 51(4) (2008), 561-569.
%H A344909 Dan Romik, <a href="https://arxiv.org/abs/1902.06330">Orthogonal polynomial expansions for the Riemann xi function.</a> arXiv:1902.06330 [math.NT], 2019.
%F A344909 T(n, k) = numerator([x^k] P(n, x), where P(n, x) = i^n*Sum_{k=0..n} (-1)^k* binomial(-3/4 + i*x, k)*binomial(-3/4 - i*x, n - k). The polynomials have the recurrence P(n, x) = (1/n)*(2*x*P(n - 1, x) - (n - 1/2)*P(n - 2, x))), starting with P(0, x) = 1 and P(1, x) = 2*x.
%e A344909 Triangle starts:
%e A344909 [0]    1;
%e A344909 [1]    0,      2;
%e A344909 [2]   -3,      0,      2;
%e A344909 [3]    0,    -13,      0,       4;
%e A344909 [4]   21,      0,    -17,       0,    2;
%e A344909 [5]    0,    177,      0,      -7,    0,    4;
%e A344909 [6]  -77,      0,   2401,       0,  -25,    0,   4;
%e A344909 [7]    0,  -4987,      0,    1123,    0,  -29,   0,   8;
%e A344909 [8] 1155,      0, -24749,       0, 1499,    0, -11,   0, 2;
%e A344909 [9]    0, 718657,      0, -341521,    0, 1157,   0, -74, 0, 4.
%p A344909 gf := ((1 - I*z)/(1 + I*z))^(I*x)*(1 + z^2)^(-3/4):
%p A344909 serz := series(gf, z, 22): coeffz := n -> coeff(serz, z, n):
%p A344909 row := n -> seq(numer(coeff(coeffz(n), x, k)), k = 0..n):
%p A344909 seq(row(n), n = 0..10);
%p A344909 # Alternative:
%p A344909 CoeffList := p -> numer(PolynomialTools:-CoefficientList(p, x)):
%p A344909 P := proc(n) option remember; if n = 0 then 1 elif n = 1 then 2*x else
%p A344909 expand((1/n)*(2*x*P(n - 1, x) - (n - 1/2)*P(n - 2, x))) fi end:
%p A344909 ListTools:-Flatten([seq(CoeffList(P(n)), n = 0..10)]);
%t A344909 ForceSimpl[a_] := Collect[Expand[Simplify[FunctionExpand[a]]], x]
%t A344909 f[n_] := I^n Sum[(-1)^k Binomial[-3/4 + I*x, k] Binomial[-3/4 - I*x, n-k], {k, 0, n}] // ForceSimpl;
%t A344909 row[n_] := CoefficientList[f[n], x] // Numerator;
%t A344909 Table[row[n], {n, 0, 10}] // Flatten
%Y A344909 Cf. A344910 (denominators).
%Y A344909 Cf. A067002 (column 0), A001316 (column n).
%K A344909 sign,tabl,frac
%O A344909 0,3
%A A344909 _Peter Luschny_, Jul 08 2021