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%I #33 Nov 02 2021 21:27:35
%S 1,0,1,0,1,1,0,1,2,1,0,2,3,3,2,0,2,5,6,5,2,0,5,6,13,13,6,5,0,10,10,16,
%T 32,16,10,10,0,28,26,36,51,51,36,26,28,0,24,50,62,74,76,74,62,50,24,0,
%U 50,50,134,138,161,161,138,134,50,50,0,124,120,146,302,345,386,345,302,146,120,124
%N Number T(n,k) of non-averaging permutations of [n] with first element k; triangle T(n,k), n >= 0, k = 0..n, read by rows.
%C A non-averaging permutation avoids any 3-term arithmetic progression.
%C T(0,0) = 1 by convention.
%H Alois P. Heinz, <a href="/A296529/b296529.txt">Rows n = 0..99, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NonaveragingSequence.html">Nonaveraging Sequence</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>
%H <a href="/index/No#non_averaging">Index entries related to non-averaging sequences</a>
%F T(n,k) = T(n,n+1-k) > 0 for k=1..n.
%e T(5,1) = 2: 15324, 15342.
%e T(5,2) = 5: 21453, 24153, 24315, 24351, 24513.
%e T(5,3) = 6: 31254, 31524, 31542, 35124, 35142, 35412.
%e T(5,4) = 5: 42153, 42315, 42351, 42513, 45213.
%e T(5,5) = 2: 51324, 51342.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 1, 2, 1;
%e 0, 2, 3, 3, 2;
%e 0, 2, 5, 6, 5, 2;
%e 0, 5, 6, 13, 13, 6, 5;
%e 0, 10, 10, 16, 32, 16, 10, 10;
%e 0, 28, 26, 36, 51, 51, 36, 26, 28;
%e 0, 24, 50, 62, 74, 76, 74, 62, 50, 24;
%e 0, 50, 50, 134, 138, 161, 161, 138, 134, 50, 50;
%e ...
%p b:= proc(s) option remember; local n, r, ok, i, j, k;
%p if nops(s) = 1 then 1
%p else n, r:= max(s), 0;
%p for j in s minus {n} do ok, i, k:= true, j-1, j+1;
%p while ok and i>=0 and k<n do ok, i, k:=
%p not i in s xor k in s, i-1, k+1 od;
%p r:= r+ `if`(ok, b(s minus {j}), 0)
%p od; r
%p fi
%p end:
%p T:= (n, k)-> `if`(k=0, 0^n, b({$0..n} minus {k-1})):
%p seq(seq(T(n, k), k=0..n), n=0..14);
%t b[s_List] := b[s] = Module[{n = Max[s], r = 0, ok, i, j, k}, If[Length[s] == 1, 1, Do[{ok, i, k} = {True, j-1, j+1}; While[ok && i >= 0 && k < n, {ok, i, k} = {FreeQ[s, i] ~Xor~ MemberQ[s, k], i-1, k+1}]; r = r + If[ok, b[s ~Complement~ {j}], 0], {j, s ~Complement~ {n}}]; r]];
%t T[0, 0]=1; T[n_, k_] := If[k==0, 0^n, b[Range[0, n] ~Complement~ {k-1}]];
%t Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 18 2017, after _Alois P. Heinz_ *)
%Y Columns k=0-1 give: A000007, A296530 (for n>0).
%Y Row sums give A003407.
%Y T(n,n) gives A296530.
%Y T(n,ceiling(n/2)) gives A296531.
%Y Cf. A292523.
%K nonn,tabl,look
%O 0,9
%A _Alois P. Heinz_, Dec 14 2017
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