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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.

%I A318254 #17 Mar 13 2020 17:00:09
%S A318254 1,1,1,1,3,-2,1,5,-20,16,1,7,-70,336,-272,1,9,-168,2016,-9792,7936,1,
%T A318254 11,-330,7392,-89760,436480,-353792,1,13,-572,20592,-466752,5674240,
%U A318254 -27595776,22368256,1,15,-910,48048,-1750320,39719680,-482926080,2348666880,-1903757312
%N A318254 Associated Omega numbers of order 2, triangle T(n,k) read by rows for n >= 0 and 0 <= k <= n.
%C A318254 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.
%F A318254 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.
%e A318254 Triangle starts:
%e A318254 [0] [1]
%e A318254 [1] [1,  1]
%e A318254 [2] [1,  3,   -2]
%e A318254 [3] [1,  5,  -20,    16]
%e A318254 [4] [1,  7,  -70,   336,    -272]
%e A318254 [5] [1,  9, -168,  2016,   -9792,    7936]
%e A318254 [6] [1, 11, -330,  7392,  -89760,  436480,   -353792]
%e A318254 [7] [1, 13, -572, 20592, -466752, 5674240, -27595776, 22368256]
%p A318254 # The function TNum is defined in A318253.
%p A318254 T := (m, n, k) -> `if`(k=0, 1, binomial(m*n-1, m*(n-k))*TNum(m, k)):
%p A318254 for n from 0 to 6 do seq(T(2, n, k), k=0..n) od;
%o A318254 (Sage)
%o A318254 def AssociatedOmegaNumberTriangle(m, len):
%o A318254     R = ZZ[x]; B = [1]*len; L = [R(1)]*len; T = [[1]]
%o A318254     for k in (1..len-1):
%o A318254         s = x*sum(binomial(m*k-1, m*(k-j))*B[j]*L[k-j] for j in (1..k-1))
%o A318254         B[k] = c = 1 - s.subs(x=1); L[k] = R(expand(s + c*x))
%o A318254         T.append([1] + [binomial(m*k-1, m*(k-j))*B[j] for j in (1..k)])
%o A318254     return T
%o A318254 A318254Triangle = lambda dim: AssociatedOmegaNumberTriangle(2, dim)
%o A318254 print(A318254Triangle(8))
%Y A318254 Even-indexed rows of A220901 (up to signs).
%Y A318254 T(n, 0) = A005408, T(n, n) = A220901 (up to signs), row sums are A040000.
%Y A318254 Cf. A318146, A318253, A318255 (m=3).
%K A318254 sign,tabl
%O A318254 0,5
%A A318254 _Peter Luschny_, Aug 26 2018