A317446 Number of permutations of [n] whose lengths of increasing runs are distinct triangular numbers.

1, 1, 0, 1, 6, 0, 1, 12, 0, 166, 3687, 20, 0, 570, 18514, 1, 16044, 689458, 1630, 46150176, 2799527248, 108527, 6182180, 0, 653209572, 50529806020, 457774882, 592018, 64091958837, 5934158290988, 7151183666, 15132424235658, 1574449800015044, 0, 342747690810188908

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..160

Formula

a(n) = 0 <=> n in { A053614 }.

a(n) > 0 <=> n in { A061208 }.

Maple

g:= (n, s)-> `if`(n in s or not issqr(8*n+1), 0, 1):
b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
      `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
       , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
    end:
a:= n-> b(n, 0$2, {}):
seq(a(n), n=0..40);

Mathematica

g[n_, s_] := If[MemberQ[s, n] || !IntegerQ@Sqrt[8*n + 1], 0, 1];
b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
     If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
     {j, 1, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]];
a[n_] := b[n, 0, 0, {}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz *)

Crossrefs

Cf. A000217, A053614, A061208, A317130, A317444, A317445, A317447, A317448.

Sequence in context: A320906 A195445 A215080 * A137943 A202189 A202183

Adjacent sequences: A317443 A317444 A317445 * A317447 A317448 A317449

Keyword

nonn

Author

Alois P. Heinz, Jul 28 2018

Status

approved