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A293368
Number of partitions 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 partition.
2
47, 544, 4232, 25100, 136516, 666800, 3142884, 14024256, 61637303, 262474700, 1109010890, 4603058016, 19018730793, 77751623552, 317106002688, 1284961711836, 5199893190893, 20961427995916, 84431958561230, 339292817869492, 1362880886322817, 5466605564267372
OFFSET
4,1
LINKS
FORMULA
a(n) ~ c * 4^n, where c = 4.90673361196637084263021203165784685586076564592828337755053385514766785... - Vaclav Kotesovec, Oct 11 2017
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(4):
seq(a(n), n=4..30);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k] Binomial[i + k - 1, k - 1]]]];
a[n_] := With[{k = 4}, Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[4, 30] (* Jean-François Alcover, Dec 12 2020, after Alois P. Heinz *)
CROSSREFS
Column k=4 of A261719.
Sequence in context: A009086 A034779 A201090 * A116146 A142253 A142577
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 07 2017
STATUS
approved