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

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

Author

Peter Luschny, Aug 26 2018

Keywords

• sign
• tabl

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.

Examples

			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]
		

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).

Programs

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;

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))

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.