A293582 - OEIS

%I #12 Dec 29 2020 10:49:15

%S 541,13060,195020,2327960,24418640,235804122,2152586500,18883155160,

%T 160908360260,1341800118020,11007289244964,89168468504160,

%U 715330888641680,5694960569676240,45067846839572000,354959016901129928,2785141532606257120,21787375678321712160

%N 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.

%H Alois P. Heinz, <a href="/A293582/b293582.txt">Table of n, a(n) for n = 5..1000</a>

%F 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

%F 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

%p b:= proc(n, k) option remember; `if`(n=0, 1,

%p       add(b(n-j, k)*binomial(j+k-1, k-1), j=1..n))

%p     end:

%p a:= n-> (k->add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(5):

%p seq(a(n), n=5..30);

%t b[n_, k_] := b[n, k] = If[n == 0, 1,

%t     Sum[b[n - j, k] Binomial[j + k - 1, k - 1], {j, 1, n}]];

%t a[n_] := With[{k = 5}, Sum[b[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];

%t a /@ Range[5, 30] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)

%Y Column k=5 of A261781.

%K nonn

%O 5,1

%A _Alois P. Heinz_, Oct 12 2017