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%I #6 Nov 26 2021 06:07:32
%S 1,-1,1,-1,0,1,0,-1,0,1,-1,0,0,0,1,1,-2,-1,1,0,1,-3,2,0,-2,2,0,1,6,-7,
%T -3,3,-3,3,0,1,-15,14,3,-10,7,-4,4,0,1,36,-37,-12,19,-19,12,-5,5,0,1,
%U -91,90,24,-54,42,-30,18,-6,6,0,1,232,-233,-67,127,-115,73,-43,25,-7,7
%N Matrix inverse of A174294.
%C First column is a signed version of A099323 with an additional leading 1.
%C First 5 rows as in A054525.
%H G. C. Greubel, <a href="/A174295/b174295.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Sum_{k=0..n} T(n, k) = A000007(n).
%F T(n, 0) = A174297(n).
%e Table begins:
%e n\k|...0...1...2...3...4...5...6...7...8...9..10
%e ---|--------------------------------------------
%e 0..|...1
%e 1..|..-1...1
%e 2..|..-1...0...1
%e 3..|...0..-1...0...1
%e 4..|..-1...0...0...0...1
%e 5..|...1..-2..-1...1...0...1
%e 6..|..-3...2...0..-2...2...0...1
%e 7..|...6..-7..-3...3..-3...3...0...1
%e 8..|.-15..14...3.-10...7..-4...4...0...1
%e 9..|..36.-37.-12..19.-19..12..-5...5...0...1
%e 10.|.-91..90..24.-54..42.-30..18..-6...6...0...1
%t t[n_, k_]:= t[n, k]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, If[k==1, Mod[n, 2], t[n-1, k-1] +t[n-2, k-1] -t[n-1, k] -t[n-2, k] ]]]; (* t = A174294 *)
%t M:= With[{m=30}, Table[t[n, k], {n,0,m}, {k,0,m}]];
%t T:= Inverse[M];
%t Table[T[[n+1, k+1]], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 25 2021 *)
%Y Cf. A000007 (row sums), A054525, A099323, A112467, A112468, A174294, A174296, A174297.
%K sign,tabl
%O 0,17
%A _Mats Granvik_, Mar 15 2010
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