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%I #11 Jul 14 2021 04:27:17
%S 1,1,0,1,6,0,1,12,0,166,3687,20,0,570,18514,1,16044,689458,1630,
%T 46150176,2799527248,108527,6182180,0,653209572,50529806020,457774882,
%U 592018,64091958837,5934158290988,7151183666,15132424235658,1574449800015044,0,342747690810188908
%N Number of permutations of [n] whose lengths of increasing runs are distinct triangular numbers.
%H Alois P. Heinz, <a href="/A317446/b317446.txt">Table of n, a(n) for n = 0..160</a>
%F a(n) = 0 <=> n in { A053614 }.
%F a(n) > 0 <=> n in { A061208 }.
%p g:= (n, s)-> `if`(n in s or not issqr(8*n+1), 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..40);
%t g[n_, s_] := If[MemberQ[s, n] || !IntegerQ@Sqrt[8*n + 1], 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, 40}] (* _Jean-François Alcover_, Jul 14 2021, after _Alois P. Heinz_ *)
%Y Cf. A000217, A053614, A061208, A317130, A317444, A317445, A317447, A317448.
%K nonn
%O 0,5
%A _Alois P. Heinz_, Jul 28 2018
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