A293582 Number of compositions of n where each part i is marked with a word of length i over a quinary alphabet whose letters appear in alphabetical order and all five letters occur at least once in the composition.

541, 13060, 195020, 2327960, 24418640, 235804122, 2152586500, 18883155160, 160908360260, 1341800118020, 11007289244964, 89168468504160, 715330888641680, 5694960569676240, 45067846839572000, 354959016901129928, 2785141532606257120, 21787375678321712160

Offset: 5

Author

Alois P. Heinz, Oct 12 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..1000

Crossrefs

Column k=5 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))(5):
seq(a(n), n=5..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 = 5}, Sum[b[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[5, 30] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

Formula

a(n) = 30*a(n-1) - 380*a(n-2) + 2690*a(n-3) - 11944*a(n-4) + 35618*a(n-5) - 74912*a(n-6) + 115104*a(n-7) - 132120*a(n-8) + 114500*a(n-9) - 74888*a(n - 10) + 36504*a(n - 11) - 12888*a(n - 12) + 3120*a(n - 13) - 464*a(n - 14) + 32*a(n - 15). - Vaclav Kotesovec, Oct 14 2017

a(n) ~ c * d^n, where d = 7.72502395887257562679242875427350515911685429396536... is the real root of the equation -2 + 10*d - 20*d^2 + 20*d^3 - 10*d^4 + d^5 = 0 and c = 0.67250239588725756267924287542735051591168542939653... is the real root of the equation -1 - 50*c - 1000*c^2 - 10000*c^3 - 50000*c^4 + 100000*c^5 = 0. - Vaclav Kotesovec, Oct 15 2017