login
A318254
Associated Omega numbers of order 2, triangle T(n,k) read by rows for n >= 0 and 0 <= k <= n.
3
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
0,5
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
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).
Sequence in context: A105954 A144252 A248033 * A002130 A089145 A324644
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Aug 26 2018
STATUS
approved