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A293368
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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.
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2
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47, 544, 4232, 25100, 136516, 666800, 3142884, 14024256, 61637303, 262474700, 1109010890, 4603058016, 19018730793, 77751623552, 317106002688, 1284961711836, 5199893190893, 20961427995916, 84431958561230, 339292817869492, 1362880886322817, 5466605564267372
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OFFSET
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4,1
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LINKS
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FORMULA
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a(n) ~ c * 4^n, where c = 4.90673361196637084263021203165784685586076564592828337755053385514766785... - Vaclav Kotesovec, Oct 11 2017
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MAPLE
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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);
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MATHEMATICA
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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}]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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