%I #16 Jul 08 2021 14:16:26

%S 1,1,1,4,1,1,1,6,1,3,32,1,6,1,3,1,80,1,3,1,15,128,1,720,1,18,1,45,1,

%T 2240,1,360,1,45,1,315,2048,1,6720,1,720,1,45,1,315,1,322560,1,90720,

%U 1,1080,1,945,1,2835,8192,1,1612800,1,181440,1,5400,1,1890,1,14175

%N T(n, k) = denominator([x^k] [z^n] ((1 - i*z)/(1 + i*z))^(i*x)*(1 + z^2)^(-3/4)). Denominators 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 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.)

%F T(n, k) = denominator([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 Triangle starts:

%e [0] 1;

%e [1] 1, 1;

%e [2] 4, 1, 1;

%e [3] 1, 6, 1, 3;

%e [4] 32, 1, 6, 1, 3;

%e [5] 1, 80, 1, 3, 1, 15;

%e [6] 128, 1, 720, 1, 18, 1, 45;

%e [7] 1, 2240, 1, 360, 1, 45, 1, 315;

%e [8] 2048, 1, 6720, 1, 720, 1, 45, 1, 315;

%e [9] 1, 322560, 1, 90720, 1, 1080, 1, 945, 1, 2835.

%p gf := ((1 - I*z)/(1 + I*z))^(I*x)*(1 + z^2)^(-3/4):

%p serz := series(gf, z, 22): coeffz := n -> coeff(serz, z, n):

%p row := n -> seq(denom(coeff(coeffz(n), x, k)), k = 0..n):

%p seq(row(n), n = 0..10);

%p # Alternative:

%p CoeffList := p -> denom(PolynomialTools:-CoefficientList(p, x)):

%p P := proc(n) option remember; if n = 0 then 1 elif n = 1 then 2*x else

%p expand((1/n)*(2*x*P(n - 1, x) - (n - 1/2)*P(n - 2, x))) fi end:

%p ListTools:-Flatten([seq(CoeffList(P(n)), n = 0..10)]);

%t ForceSimpl[a_] := Collect[Expand[Simplify[FunctionExpand[a]]], x]

%t 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 row[n_] := CoefficientList[f[n], x] // Denominator;

%t Table[row[n], {n, 0, 10}] // Flatten

%Y Cf. A344909 (numerators).

%Y Cf. A088802 and A123854 (denominator(binomial(1/4, n)) for column 0.

%Y Cf. A049606 (numerator(n!/2^n)) for column n.

%K nonn,tabl,frac

%O 0,4

%A _Peter Luschny_, Jul 08 2021