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%I #24 Mar 06 2022 08:41:38
%S 1,1,2,3,3,24,0,100,15,0,5,594,108,18,0,4389,504,119,21,0,7,35744,
%T 3520,960,64,32,0,325395,31077,5238,927,207,27,0,9,3288600,288300,
%U 42050,8800,900,100,50,0,36489992,2946141,409827,59785,9174,1518,319,33,0,11
%N Triangle read by rows: T(n,k) is the number of permutations of [1..n] with k modular progressions of rise 2, distance 1 and length 3 (n >= 0, k >= 0).
%D Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, Congressus Numerantium, Vol. 208 (2011), pp. 147-165.
%H Alois P. Heinz, <a href="/A216724/b216724.txt">Rows n = 0..18, flattened</a>
%e Triangle begins:
%e 1
%e 1
%e 2
%e 3 3
%e 24 0
%e 100 15 0 5
%e 594 108 18 0
%e 4389 504 119 21 0 7
%e 35744 3520 960 64 32 0
%e 325395 31077 5238 927 207 27 0 9
%e 3288600 288300 42050 8800 900 100 50 0
%e ...
%p b:= proc(s, x, y, n) option remember; expand(`if`(s={}, 1, add(
%p `if`(x>0 and irem(n+x-y, n)=2 and irem(n+y-j, n)=2, z, 1)*
%p b(s minus {j}, y, j, n), j=s)))
%p end:
%p T:= n-> (p-> seq(coeff(p, z, i), i=0..max(0,
%p iquo(n-1,2)*2-1)))(b({$1..n}, 0$2, n)):
%p seq(T(n), n=0..11); # _Alois P. Heinz_, Apr 13 2021
%t b[s_, x_, y_, n_] := b[s, x, y, n] = Expand[If[s == {}, 1, Sum[
%t If[x>0 && Mod[n + x - y, n] == 2 && Mod[n + y - j, n] == 2, z, 1]*
%t b[s~Complement~{j}, y, j, n], {j, s}]]];
%t T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Max[0,
%t Quotient[n - 1, 2]*2 - 1]}]][b[Range[n], 0, 0, n]];
%t Table[T[n], {n, 0, 11}] // Flatten (* _Jean-François Alcover_, Mar 06 2022, after _Alois P. Heinz_ *)
%Y Column 1 is A174073.
%Y Row sums are A000142.
%Y Cf. A216716, A216718, A216719, A216722.
%K nonn,tabf
%O 0,3
%A _N. J. A. Sloane_, Sep 15 2012
%E More terms from _Alois P. Heinz_, Apr 13 2021
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