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

Terms

a(0) =1a(1) =1a(2) =0a(3) =1a(4) =6a(5) =0a(6) =1a(7) =12a(8) =0a(9) =166a(10) =3687a(11) =20a(12) =0a(13) =570a(14) =18514a(15) =1a(16) =16044a(17) =689458a(18) =1630a(19) =46150176a(20) =2799527248a(21) =108527a(22) =6182180a(23) =0a(24) =653209572a(25) =50529806020a(26) =457774882a(27) =592018a(28) =64091958837a(30) =7151183666