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A344910 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.

%I A344910 #22 May 30 2025 02:15:10
%S A344910 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 A344910 2240,1,360,1,45,1,315,2048,1,6720,1,720,1,45,1,315,1,322560,1,90720,
%U A344910 1,1080,1,945,1,2835,8192,1,1612800,1,181440,1,5400,1,1890,1,14175
%N A344910 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 A344910 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; pp. 213-216.
%F A344910 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 A344910 Triangle starts:
%e A344910   [0] 1;
%e A344910   [1] 1, 1;
%e A344910   [2] 4, 1, 1;
%e A344910   [3] 1, 6, 1, 3;
%e A344910   [4] 32, 1, 6, 1, 3;
%e A344910   [5] 1, 80, 1, 3, 1, 15;
%e A344910   [6] 128, 1, 720, 1, 18, 1, 45;
%e A344910   [7] 1, 2240, 1, 360, 1, 45, 1, 315;
%e A344910   [8] 2048, 1, 6720, 1, 720, 1, 45, 1, 315;
%e A344910   [9] 1, 322560, 1, 90720, 1, 1080, 1, 945, 1, 2835.
%p A344910 gf := ((1 - I*z)/(1 + I*z))^(I*x)*(1 + z^2)^(-3/4):
%p A344910 serz := series(gf, z, 22): coeffz := n -> coeff(serz, z, n):
%p A344910 row := n -> seq(denom(coeff(coeffz(n), x, k)), k = 0..n):
%p A344910 seq(row(n), n = 0..10);
%p A344910 # Alternative:
%p A344910 CoeffList := p -> denom(PolynomialTools:-CoefficientList(p, x)):
%p A344910 P := proc(n) option remember; if n = 0 then 1 elif n = 1 then 2*x else
%p A344910 expand((1/n)*(2*x*P(n - 1, x) - (n - 1/2)*P(n - 2, x))) fi end:
%p A344910 ListTools:-Flatten([seq(CoeffList(P(n)), n = 0..10)]);
%t A344910 ForceSimpl[a_] := Collect[Expand[Simplify[FunctionExpand[a]]], x]
%t A344910 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 A344910 row[n_] := CoefficientList[f[n], x] // Denominator;
%t A344910 Table[row[n], {n, 0, 10}] // Flatten
%Y A344910 Cf. A344909 (numerators).
%Y A344910 Cf. A088802 and A123854 (denominator(binomial(1/4, n))) for column 0.
%Y A344910 Cf. A049606 (numerator(n!/2^n)) for column n.
%K A344910 nonn,tabl,frac
%O A344910 0,4
%A A344910 _Peter Luschny_, Jul 08 2021