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

%I A318146 #37 Mar 06 2020 03:29:53
%S A318146 1,0,1,0,-2,3,0,16,-30,15,0,-272,588,-420,105,0,7936,-18960,16380,
%T A318146 -6300,945,0,-353792,911328,-893640,429660,-103950,10395,0,22368256,
%U A318146 -61152000,65825760,-36636600,11351340,-1891890,135135
%N A318146 Coefficients of the Omega polynomials of order 2, triangle T(n,k) read by rows with 0<=k<=n.
%C A318146 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.
%C A318146 The signed Euler secant numbers appear as values at x=-1 and the signed Euler tangent numbers as the coefficients of x.
%H A318146 Peter Luschny, <a href="/A318146/b318146.txt">Table of the first 63 rows.</a>
%F A318146 OmegaPolynomial(m, n) = (m*n)! [z^n] E(m, z)^x where E(m, z) is the Mittag-Leffler function.
%F A318146 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).
%F A318146 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.
%F A318146 T(n, k) = [x^k] (2*n)! [z^(2*n)] sech(z)^(-x). - _Peter Luschny_, Jul 01 2019
%e A318146 Row n in the triangle below is the coefficient list of OmegaPolynomial(2, n). For other cases than m = 2 see the cross-references.
%e A318146 [0] [1]
%e A318146 [1] [0,        1]
%e A318146 [2] [0,       -2,         3]
%e A318146 [3] [0,       16,       -30,       15]
%e A318146 [4] [0,     -272,       588,     -420,       105]
%e A318146 [5] [0,     7936,    -18960,    16380,     -6300,      945]
%e A318146 [6] [0,  -353792,    911328,  -893640,    429660,  -103950,    10395]
%e A318146 [7] [0, 22368256, -61152000, 65825760, -36636600, 11351340, -1891890, 135135]
%p A318146 OmegaPolynomial := proc(m, n) local Omega;
%p A318146 Omega := m -> hypergeom([], [seq(i/m, i=1..m-1)], (z/m)^m):
%p A318146 series(Omega(m)^x, z, m*(n+1)):
%p A318146 sort(expand((m*n)!*coeff(%, z, n*m)), [x], ascending) end:
%p A318146 CL := p -> PolynomialTools:-CoefficientList(p, x):
%p A318146 FL := p -> ListTools:-Flatten(p):
%p A318146 FL([seq(CL(OmegaPolynomial(2, n)), n=0..8)]);
%p A318146 # Alternative:
%p A318146 ser := series(sech(z)^(-x), z, 24): row := n -> n!*coeff(ser, z, n):
%p A318146 seq(seq(coeff(row(2*n), x, k), k=0..n), n=0..6); # _Peter Luschny_, Jul 01 2019
%t A318146 OmegaPolynomial[m_,n_] :=  Module [{ },
%t A318146 S = Series[MittagLefflerE[m,z]^x, {z,0,10}];
%t A318146 Expand[(m n)! Coefficient[S,z,n]] ]
%t A318146 Table[CoefficientList[OmegaPolynomial[2,n],x], {n,0,7}] // Flatten
%t A318146 (* Second program: *)
%t A318146 T[n_, k_] := (2n)! SeriesCoefficient[Sech[z]^-x, {z, 0, 2n}, {x, 0, k}];
%t A318146 Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 23 2019, after _Peter Luschny_ *)
%o A318146 (Sage)
%o A318146 def OmegaPolynomial(m, n):
%o A318146     R = ZZ[x]; z = var('z')
%o A318146     f = [i/m for i in (1..m-1)]
%o A318146     h = lambda z: hypergeometric([], f, (z/m)^m)
%o A318146     return R(factorial(m*n)*taylor(h(z)^x, z, 0, m*n + 1).coefficient(z, m*n))
%o A318146 [list(OmegaPolynomial(2, n)) for n in (0..6)]
%o A318146 # Recursion over the polynomials, returns a list of the first len polynomials:
%o A318146 def OmegaPolynomials(m, len, coeffs=true):
%o A318146     R = ZZ[x]; B = [0]*len; L = [R(1)]*len
%o A318146     for k in (1..len-1):
%o A318146         s = x*sum(binomial(m*k-1, m*(k-j))*B[j]*L[k-j] for j in (1..k-1))
%o A318146         B[k] = c = 1 - s.subs(x=1)
%o A318146         L[k] = R(expand(s + c*x))
%o A318146     return [list(l) for l in L] if coeffs else L
%o A318146 print(OmegaPolynomials(2, 6))
%Y A318146 Variant is A088874 (unsigned).
%Y A318146 T(n,1) = A000182(n), T(n,n) = A001147(n).
%Y A318146 All row sums are 1, alternating row sums are A028296 (A000364).
%Y A318146 A023531 (m=1), this seq (m=2), A318147 (m=3), A318148 (m=4).
%Y A318146 Associated Omega numbers: A318254 (m=2), A318255 (m=3).
%Y A318146 Coefficients of x for Omega polynomials of all orders are in A318253.
%K A318146 sign,tabl
%O A318146 0,5
%A A318146 _Peter Luschny_, Aug 22 2018