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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.
2
75, 1232, 13064, 114032, 893490, 6550112, 45966744, 313094512, 2088274012, 13719804224, 89151186688, 574612403008, 3681207840264, 23476261805376, 149202047915680, 945775992492352, 5983286739107952, 37794913734696448, 238464380911582336, 1503238554666345728
OFFSET
4,1
LINKS
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 12 2017
STATUS
approved