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

5,1

Links

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

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

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 *)

Crossrefs

Column k=5 of A261781.

Sequence in context: A226800 A320619 A218095 * A331642 A263065 A325217

Adjacent sequences: A293579 A293580 A293581 * A293583 A293584 A293585

Keyword

nonn

Author

Alois P. Heinz, Oct 12 2017

Status

approved