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%I #13 Jul 27 2021 07:50:38
%S 1,1,0,0,1,8,0,0,0,1,18,0,0,1428,47998,0,1,32,0,0,9688,505056,0,0,0,
%T 4085949,284958912,0,0,290824632172,28643427712626,0,0,0,104902510,
%U 9998016202,1,72,23207824626842,3008268832634364,182778,206173972520,24290829974718,0
%N Number of permutations of [n] whose lengths of increasing runs are distinct squares.
%H Alois P. Heinz, <a href="/A317445/b317445.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = 0 <=> n in { A001422 }.
%F a(n) > 0 <=> n in { A003995 }.
%p g:= (n, s)-> `if`(n in s or not issqr(n), 0, 1):
%p b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
%p `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
%p , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
%p end:
%p a:= n-> b(n, 0$2, {}):
%p seq(a(n), n=0..50);
%t g[n_, s_] := If[MemberQ[s, n] || !IntegerQ@Sqrt[n], 0, 1];
%t b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
%t If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
%t {j, 1, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]];
%t a[n_] := b[n, 0, 0, {}];
%t Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jul 24 2021, after _Alois P. Heinz_ *)
%Y Cf. A000290, A001422, A003995, A317129, A317444, A317446, A317447, A317448.
%K nonn
%O 0,6
%A _Alois P. Heinz_, Jul 28 2018
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