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A318254 Associated Omega numbers of order 2, triangle T(n,k) read by rows for n >= 0 and 0 <= k <= n. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=0..44.

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

Adjacent sequences:  A318251 A318252 A318253 * A318255 A318256 A318257

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Aug 26 2018

STATUS

approved

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Last modified December 8 02:32 EST 2021. Contains 349590 sequences. (Running on oeis4.)