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%I #29 Jan 02 2023 12:30:48
%S 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,
%T 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,
%U 1,1,2,3,5,9,13,17,14,1,0,1,1,1,2,3,5,9,14,22,28,21,1,0
%N 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- ... ))).
%H Alois P. Heinz, <a href="/A185646/b185646.txt">Antidiagonals n = 0..140, flattened</a>
%H Paul D. Hanna et al., <a href="http://list.seqfan.eu/oldermail/seqfan/2013-July/011445.html">Formula Needed for a Family of Continued Fractions</a> and follow-up messages on the SeqFan list, Jul 28 2013
%e Square array A(n,m) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 2, 2, 2, 2, 2, 2, ...
%e 0, 1, 2, 3, 3, 3, 3, 3, 3, ...
%e -1, 1, 3, 4, 5, 5, 5, 5, 5, ...
%e 0, 1, 5, 7, 8, 9, 9, 9, 9, ...
%e 0, 1, 6, 11, 13, 14, 15, 15, 15, ...
%e -1, 1, 10, 17, 22, 24, 25, 26, 26, ...
%t 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 *)
%Y Columns m=0-10 give: A143064, A000012, A227360, A173173(n+1), A227374, A227375, A228646, A228644, A185648, A228645, A185649.
%Y Diagonal gives: A005169.
%K sign,tabl
%O 0,19
%A _Alois P. Heinz_, Aug 29 2013
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