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A144432 Triangle, T(n, k), read by rows: T(n, k) = t(n, k)^2 - t(n, k) - 1, where t(n,k) = (m*(n-k) + 1)*t(n-1, k-1) + (m*k - (m-1))*t(n-1, k) and m = -1.

%I A144432 #11 Mar 02 2022 08:57:25
%S A144432 -1,-1,-1,-1,-1,-1,-1,1,1,-1,-1,5,1,5,-1,-1,11,1,1,11,-1,-1,19,41,71,
%T A144432 41,19,-1,-1,29,71,29,29,71,29,-1,-1,41,239,701,869,701,239,41,-1,-1,
%U A144432 55,379,811,181,181,811,379,55,-1
%N A144432 Triangle, T(n, k), read by rows: T(n, k) = t(n, k)^2 - t(n, k) - 1, where t(n,k) = (m*(n-k) + 1)*t(n-1, k-1) + (m*k - (m-1))*t(n-1, k) and m = -1.
%H A144432 G. C. Greubel, <a href="/A144432/b144432.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A144432 T(n, k) = t(n, k)^2 - t(n, k) - 1, where t(n,k) = (m*(n-k) + 1)*t(n-1, k-1) + (m*k - (m-1))*t(n-1, k) and m = -1.
%F A144432 From _G. C. Greubel_, Mar 02 2022: (Start)
%F A144432 T(n, n-k) = T(n, k).
%F A144432 T(n, k) = t(n,k)^2 - t(n,k) - 1, where t(n,k) = (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3) with t(1, k) = t(2, k) = 1.
%F A144432 Sum_{k=1..n} T(n,k) = -n*[n<4] + ( 2*binomial(2*n-6, n-3)*(binomial(n-1,2) - (-1)^n*binomial(n-3,2))/binomial(n-1,2) - n )*[n>=4]. (End)
%e A144432 Triangle begins as:
%e A144432   -1;
%e A144432   -1, -1;
%e A144432   -1, -1,  -1;
%e A144432   -1,  1,   1,  -1;
%e A144432   -1,  5,   1,   5,  -1;
%e A144432   -1, 11,   1,   1,  11,  -1;
%e A144432   -1, 19,  41,  71,  41,  19,  -1;
%e A144432   -1, 29,  71,  29,  29,  71,  29,  -1;
%e A144432   -1, 41, 239, 701, 869, 701, 239,  41, -1;
%e A144432   -1, 55, 379, 811, 181, 181, 811, 379, 55, -1;
%t A144432 t[n_, k_, m_]:= t[n, k, m]= If[k==1 || k==n, 1, (m*(n-k)+1)*t[n-1,k-1,m] + (m*(k - 1)+1)*t[n-1,k,m]];
%t A144432 T[n_, k_, m_]:= t[n,k,m]^2 -t[n,k,m] -1;
%t A144432 Table[T[n,k,-1], {n,15}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 02 2022 *)
%o A144432 (Sage)
%o A144432 def t(n,k):
%o A144432     if (n<3): return 1
%o A144432     else: return (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3)
%o A144432 def A144432(n,k): return t(n,k)^2 - t(n,k) - 1
%o A144432 flatten([[A144432(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 02 2022
%Y A144432 Cf. A098593, A144431.
%K A144432 sign,tabl
%O A144432 1,12
%A A144432 _Roger L. Bagula_, Oct 04 2008
%E A144432 Edited by _G. C. Greubel_, Mar 02 2022