A293367 Number of partitions of n where each part i is marked with a word of length i over a ternary alphabet whose letters appear in alphabetical order and all three letters occur at least once in the partition.

10, 81, 396, 1751, 6528, 23892, 80979, 272085, 876342, 2821217, 8840964, 27713589, 85532512, 263935014, 806417553, 2464692788, 7483544643, 22727335830, 68734242687, 207887123472, 627024671262, 1891376241178, 5694616254570, 17146333061406, 51564199968339

Offset: 3

Author

Alois P. Heinz, Oct 07 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..1000

Crossrefs

Column k=3 of A261719.

Cf. A261737.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(3):
seq(a(n), n=3..30);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k] Binomial[i + k - 1, k - 1]]]];
a[n_] := With[{k = 3}, Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[3, 30] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 3^n, where c = 6.846206073498521357898163368676070142316815386135993166380819930419737... - Vaclav Kotesovec, Oct 11 2017