A293581 Number of compositions of n where each part i is marked with a word of length i over a quaternary alphabet whose letters appear in alphabetical order and all four letters occur at least once in the composition.

75, 1232, 13064, 114032, 893490, 6550112, 45966744, 313094512, 2088274012, 13719804224, 89151186688, 574612403008, 3681207840264, 23476261805376, 149202047915680, 945775992492352, 5983286739107952, 37794913734696448, 238464380911582336, 1503238554666345728

Offset: 4

Author

Alois P. Heinz, Oct 12 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..1000

Crossrefs

Column k=4 of A261781.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
a:= n-> (k->add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(4):
seq(a(n), n=4..30);

Mathematica

b[n_, k_] := b[n, k] = If[n == 0, 1,
    Sum[b[n - j, k] Binomial[j + k - 1, k - 1], {j, 1, n}]];
a[n_] := With[{k = 4}, Sum[b[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[4, 30] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

Formula

From Vaclav Kotesovec, Oct 14 2017: (Start)

a(n) = 20*a(n-1) - 160*a(n-2) + 670*a(n-3) - 1634*a(n-4) + 2476*a(n-5) - 2432*a(n-6) + 1564*a(n-7) - 640*a(n-8) + 152*a(n-9) - 16*a(n-10).

a(n) ~ (1 + sqrt(2) + sqrt(4 + 3*sqrt(2)))/8 * (2 + sqrt(2) + sqrt(4 + 3*sqrt(2)))^n. (End)