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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 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.)