OFFSET
0,5
COMMENTS
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.
FORMULA
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.
EXAMPLE
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]
MAPLE
# 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;
PROG
(Sage)
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))
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Aug 26 2018
STATUS
approved