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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.

%I A293581 #13 Dec 29 2020 10:48:59
%S A293581 75,1232,13064,114032,893490,6550112,45966744,313094512,2088274012,
%T A293581 13719804224,89151186688,574612403008,3681207840264,23476261805376,
%U A293581 149202047915680,945775992492352,5983286739107952,37794913734696448,238464380911582336,1503238554666345728
%N 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.
%H A293581 Alois P. Heinz, <a href="/A293581/b293581.txt">Table of n, a(n) for n = 4..1000</a>
%F A293581 From _Vaclav Kotesovec_, Oct 14 2017: (Start)
%F A293581 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).
%F A293581 a(n) ~ (1 + sqrt(2) + sqrt(4 + 3*sqrt(2)))/8 * (2 + sqrt(2) + sqrt(4 + 3*sqrt(2)))^n. (End)
%p A293581 b:= proc(n, k) option remember; `if`(n=0, 1,
%p A293581       add(b(n-j, k)*binomial(j+k-1, k-1), j=1..n))
%p A293581     end:
%p A293581 a:= n-> (k->add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(4):
%p A293581 seq(a(n), n=4..30);
%t A293581 b[n_, k_] := b[n, k] = If[n == 0, 1,
%t A293581     Sum[b[n - j, k] Binomial[j + k - 1, k - 1], {j, 1, n}]];
%t A293581 a[n_] := With[{k = 4}, Sum[b[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
%t A293581 a /@ Range[4, 30] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)
%Y A293581 Column k=4 of A261781.
%K A293581 nonn
%O A293581 4,1
%A A293581 _Alois P. Heinz_, Oct 12 2017