A185646 Square array A(n,m), n>=0, m>=0, read by antidiagonals, where column m is the expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))).

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, -1, 1, 1, 1, 2, 2, 1, 0, 1, 1, 1, 2, 3, 3, 1, 0, 1, 1, 1, 2, 3, 4, 5, 1, -1, 1, 1, 1, 2, 3, 5, 7, 6, 1, 0, 1, 1, 1, 2, 3, 5, 8, 11, 10, 1, 0, 1, 1, 1, 2, 3, 5, 9, 13, 17, 14, 1, 0, 1, 1, 1, 2, 3, 5, 9, 14, 22, 28, 21, 1, 0

Offset: 0

Author

Alois P. Heinz, Aug 29 2013

Keywords

• sign
• tabl

Examples

			Square array A(n,m) begins:
   1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   0,  1,  1,  1,  1,  1,  1,  1,  1, ...
   0,  1,  2,  2,  2,  2,  2,  2,  2, ...
   0,  1,  2,  3,  3,  3,  3,  3,  3, ...
  -1,  1,  3,  4,  5,  5,  5,  5,  5, ...
   0,  1,  5,  7,  8,  9,  9,  9,  9, ...
   0,  1,  6, 11, 13, 14, 15, 15, 15, ...
  -1,  1, 10, 17, 22, 24, 25, 26, 26, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013

Crossrefs

Columns m=0-10 give: A143064, A000012, A227360, A173173(n+1), A227374, A227375, A228646, A228644, A185648, A228645, A185649.

Diagonal gives: A005169.

Programs

Mathematica

nMax = 12; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A = Table[col[m][[1 ;; nMax + 1]], {m, 0, nMax}] // Transpose; a[n_ /; 0 <= n <= nMax, m_ /; 0 <= m <= nMax] := With[{n1 = n + 1, m1 = m + 1}, A[[n1, m1]]]; Table[a[n - m, m], {n, 0, nMax}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2016 *)