A318255 -id:A318255 - cp's OEIS frontend

A318146 Coefficients of the Omega polynomials of order 2, triangle T(n,k) read by rows with 0<=k<=n.

1, 0, 1, 0, -2, 3, 0, 16, -30, 15, 0, -272, 588, -420, 105, 0, 7936, -18960, 16380, -6300, 945, 0, -353792, 911328, -893640, 429660, -103950, 10395, 0, 22368256, -61152000, 65825760, -36636600, 11351340, -1891890, 135135

Offset: 0

Peter Luschny, Aug 22 2018

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.

			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]

Peter Luschny, Table of the first 63 rows.

Variant is A088874 (unsigned).
T(n,1) = A000182(n), T(n,n) = A001147(n).
All row sums are 1, alternating row sums are A028296 (A000364).
A023531 (m=1), this seq (m=2), A318147 (m=3), A318148 (m=4).
Associated Omega numbers: A318254 (m=2), A318255 (m=3).
Coefficients of x for Omega polynomials of all orders are in A318253.

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

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 *)

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))

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

A318254 Associated Omega numbers of order 2, triangle T(n,k) read by rows for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, -2, 1, 5, -20, 16, 1, 7, -70, 336, -272, 1, 9, -168, 2016, -9792, 7936, 1, 11, -330, 7392, -89760, 436480, -353792, 1, 13, -572, 20592, -466752, 5674240, -27595776, 22368256, 1, 15, -910, 48048, -1750320, 39719680, -482926080, 2348666880, -1903757312

Offset: 0

Peter Luschny, Aug 26 2018

The Omega polynomials A318146 are defined 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. 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.

			Triangle starts:
[0] [1]
[1] [1,  1]
[2] [1,  3,   -2]
[3] [1,  5,  -20,    16]
[4] [1,  7,  -70,   336,    -272]
[5] [1,  9, -168,  2016,   -9792,    7936]
[6] [1, 11, -330,  7392,  -89760,  436480,   -353792]
[7] [1, 13, -572, 20592, -466752, 5674240, -27595776, 22368256]

Even-indexed rows of A220901 (up to signs).
T(n, 0) = A005408, T(n, n) = A220901 (up to signs), row sums are A040000.
Cf. A318146, A318253, A318255 (m=3).

# The function TNum is defined in A318253.
T := (m, n, k) -> `if`(k=0, 1, binomial(m*n-1, m*(n-k))*TNum(m, k)):
for n from 0 to 6 do seq(T(2, n, k), k=0..n) od;

def AssociatedOmegaNumberTriangle(m, len):
    R = ZZ[x]; B = [1]*len; L = [R(1)]*len; T = [[1]]
    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))
        T.append([1] + [binomial(m*k-1, m*(k-j))*B[j] for j in (1..k)])
    return T
A318254Triangle = lambda dim: AssociatedOmegaNumberTriangle(2, dim)
print(A318254Triangle(8))

T(m, n, k) = binomial(m*n-1, m*(n-k))*A318253(m, k) for k>0 and 1 for k=0. We consider here the case m=2.

cp's OEIS Frontend

A318146 Coefficients of the Omega polynomials of order 2, triangle T(n,k) read by rows with 0<=k<=n.

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