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A320554
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Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most four elements and for at least one block c the smallest integer interval containing c has exactly four elements.
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3
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5, 17, 45, 121, 336, 901, 2347, 6014, 15314, 38766, 97531, 244054, 608339, 1511919, 3748379, 9273353, 22901665, 56477538, 139114445, 342325451, 841676972, 2067997764, 5078117000, 12463618356, 30577931115, 74993361731, 183870516407, 450708620604, 1104563863868
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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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G.f.: -(-x^8 -3*x^7 +x^6 +7*x^5 +5*x^4) / (x^12 +5*x^11 +7*x^10 -x^9 -13*x^8 -19*x^7 -14*x^6 -6*x^5 +x^4 +2*x^2 +2*x-1).
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EXAMPLE
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a(5) = 17: 1234|5, 124|35, 124|3|5, 134|25, 134|2|5, 13|245, 13|25|4, 14|235, 14|23|5, 1|2345, 1|235|4, 14|25|3, 14|2|35, 14|2|3|5, 1|245|3, 1|25|34, 1|25|3|4.
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MAPLE
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b:= proc(n, m, l) option remember; `if`(n=0, 1,
add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
`if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
a:= n-> (k-> A(n, k) -`if`(k=0, 0, A(n, k-1)))(4):
seq(a(n), n=4..35);
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MATHEMATICA
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b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1]~Complement~{0}}]];
A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0 &, k - 1]]]];
a[n_] := With[{k = 4}, A[n, k] - If[k == 0, 0, A[n, k - 1]]];
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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