OFFSET
0,5
COMMENTS
The name 'Omega polynomial' is not a standard name. The Omega numbers are the coefficients of the Omega polynomials, the associated Omega numbers are the weights of P(m, k) in the recurrence formula given below.
The signed Euler secant numbers appear as values at x=-1 and the signed Euler tangent numbers as the coefficients of x.
LINKS
Peter Luschny, Table of the first 63 rows.
FORMULA
OmegaPolynomial(m, n) = (m*n)! [z^n] E(m, z)^x where E(m, z) is the Mittag-Leffler function.
OmegaPolynomial(m, n) = (m*n)!*[z^(n*m)] H(m, z)^x where H(m, z) = hypergeom([], [seq(i/m, i=1..m-1)], (z/m)^m).
The Omega polynomials can be computed by the recurrence P(m, 0) = 1 and for n >= 1 P(m, n) = x * Sum_{k=0..n-1} binomial(m*n-1, m*k)*T(m, n-k)*P(m, k) where T(m, n) are the generalized tangent numbers A318253. A separate computation of the T(m, n) can be avoided, see the Sage implementation below for the details.
T(n, k) = [x^k] (2*n)! [z^(2*n)] sech(z)^(-x). - Peter Luschny, Jul 01 2019
EXAMPLE
Row n in the triangle below is the coefficient list of OmegaPolynomial(2, n). For other cases than m = 2 see the cross-references.
[0] [1]
[1] [0, 1]
[2] [0, -2, 3]
[3] [0, 16, -30, 15]
[4] [0, -272, 588, -420, 105]
[5] [0, 7936, -18960, 16380, -6300, 945]
[6] [0, -353792, 911328, -893640, 429660, -103950, 10395]
[7] [0, 22368256, -61152000, 65825760, -36636600, 11351340, -1891890, 135135]
MAPLE
OmegaPolynomial := proc(m, n) local Omega;
Omega := m -> hypergeom([], [seq(i/m, i=1..m-1)], (z/m)^m):
series(Omega(m)^x, z, m*(n+1)):
sort(expand((m*n)!*coeff(%, z, n*m)), [x], ascending) end:
CL := p -> PolynomialTools:-CoefficientList(p, x):
FL := p -> ListTools:-Flatten(p):
FL([seq(CL(OmegaPolynomial(2, n)), n=0..8)]);
# Alternative:
ser := series(sech(z)^(-x), z, 24): row := n -> n!*coeff(ser, z, n):
seq(seq(coeff(row(2*n), x, k), k=0..n), n=0..6); # Peter Luschny, Jul 01 2019
MATHEMATICA
OmegaPolynomial[m_, n_] := Module [{ },
S = Series[MittagLefflerE[m, z]^x, {z, 0, 10}];
Expand[(m n)! Coefficient[S, z, n]] ]
Table[CoefficientList[OmegaPolynomial[2, n], x], {n, 0, 7}] // Flatten
(* Second program: *)
T[n_, k_] := (2n)! SeriesCoefficient[Sech[z]^-x, {z, 0, 2n}, {x, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 23 2019, after Peter Luschny *)
PROG
(Sage)
def OmegaPolynomial(m, n):
R = ZZ[x]; z = var('z')
f = [i/m for i in (1..m-1)]
h = lambda z: hypergeometric([], f, (z/m)^m)
return R(factorial(m*n)*taylor(h(z)^x, z, 0, m*n + 1).coefficient(z, m*n))
[list(OmegaPolynomial(2, n)) for n in (0..6)]
# Recursion over the polynomials, returns a list of the first len polynomials:
def OmegaPolynomials(m, len, coeffs=true):
R = ZZ[x]; B = [0]*len; L = [R(1)]*len
for k in (1..len-1):
s = x*sum(binomial(m*k-1, m*(k-j))*B[j]*L[k-j] for j in (1..k-1))
B[k] = c = 1 - s.subs(x=1)
L[k] = R(expand(s + c*x))
return [list(l) for l in L] if coeffs else L
print(OmegaPolynomials(2, 6))
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Aug 22 2018
STATUS
approved