A174295 Matrix inverse of A174294.

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, -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, -91, 90, 24, -54, 42, -30, 18, -6, 6, 0, 1, 232, -233, -67, 127, -115, 73, -43, 25, -7, 7

Offset: 0

Author

Mats Granvik, Mar 15 2010

Keywords

• sign
• tabl

Comments

First column is a signed version of A099323 with an additional leading 1.

First 5 rows as in A054525.

Examples

			Table begins:
  n\k|...0...1...2...3...4...5...6...7...8...9..10
  ---|--------------------------------------------
  0..|...1
  1..|..-1...1
  2..|..-1...0...1
  3..|...0..-1...0...1
  4..|..-1...0...0...0...1
  5..|...1..-2..-1...1...0...1
  6..|..-3...2...0..-2...2...0...1
  7..|...6..-7..-3...3..-3...3...0...1
  8..|.-15..14...3.-10...7..-4...4...0...1
  9..|..36.-37.-12..19.-19..12..-5...5...0...1
  10.|.-91..90..24.-54..42.-30..18..-6...6...0...1
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000007 (row sums), A054525, A099323, A112467, A112468, A174294, A174296, A174297.

Programs

Mathematica

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 *)
M:= With[{m=30}, Table[t[n, k], {n,0,m}, {k,0,m}]];
T:= Inverse[M];
Table[T[[n+1, k+1]], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 25 2021 *)

Formula

Sum_{k=0..n} T(n, k) = A000007(n).

T(n, 0) = A174297(n).