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

4,1

Links

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

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)

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 *)

Crossrefs

Column k=4 of A261781.

Sequence in context: A320618 A218094 A285920 * A210047 A210505 A285856

Adjacent sequences: A293578 A293579 A293580 * A293582 A293583 A293584

Keyword

nonn

Author

Alois P. Heinz, Oct 12 2017

Status

approved