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.
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, 2240, 1, 360, 1, 45, 1, 315, 2048, 1, 6720, 1, 720, 1, 45, 1, 315, 1, 322560, 1, 90720, 1, 1080, 1, 945, 1, 2835, 8192, 1, 1612800, 1, 181440, 1, 5400, 1, 1890, 1, 14175
Offset: 0
Examples
Triangle starts: [0] 1; [1] 1, 1; [2] 4, 1, 1; [3] 1, 6, 1, 3; [4] 32, 1, 6, 1, 3; [5] 1, 80, 1, 3, 1, 15; [6] 128, 1, 720, 1, 18, 1, 45; [7] 1, 2240, 1, 360, 1, 45, 1, 315; [8] 2048, 1, 6720, 1, 720, 1, 45, 1, 315; [9] 1, 322560, 1, 90720, 1, 1080, 1, 945, 1, 2835.
Links
- R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer, 2010; pp. 213-216.
Crossrefs
Programs
-
Maple
gf := ((1 - I*z)/(1 + I*z))^(I*x)*(1 + z^2)^(-3/4): serz := series(gf, z, 22): coeffz := n -> coeff(serz, z, n): row := n -> seq(denom(coeff(coeffz(n), x, k)), k = 0..n): seq(row(n), n = 0..10); # Alternative: CoeffList := p -> denom(PolynomialTools:-CoefficientList(p, x)): P := proc(n) option remember; if n = 0 then 1 elif n = 1 then 2*x else expand((1/n)*(2*x*P(n - 1, x) - (n - 1/2)*P(n - 2, x))) fi end: ListTools:-Flatten([seq(CoeffList(P(n)), n = 0..10)]);
-
Mathematica
ForceSimpl[a_] := Collect[Expand[Simplify[FunctionExpand[a]]], x] f[n_] := I^n Sum[(-1)^k Binomial[-3/4 + I*x, k] Binomial[-3/4 - I*x, n-k], {k, 0, n}] // ForceSimpl; row[n_] := CoefficientList[f[n], x] // Denominator; Table[row[n], {n, 0, 10}] // Flatten
Formula
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.