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A327876
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Number of set partitions of [n] with distinct block sizes and one of the block sizes is 1.
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2
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0, 1, 0, 3, 4, 5, 66, 112, 456, 765, 15070, 31856, 150756, 663962, 1943046, 44316105, 127348864, 661449549, 3220447446, 20913769072, 68889553260, 2403704171190, 7894983374674, 51012052828947, 270186003789312, 1836634462055350, 13402212376611266
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OFFSET
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0,4
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COMMENTS
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Sum of multinomials M(n; lambda), where lambda ranges over all integer partitions of n into distinct parts and one part is 1.
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LINKS
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FORMULA
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a(1) = 1: 1.
a(2) = 0.
a(3) = 3: 12|3, 13|2, 1|23.
a(4) = 4: 123|4, 124|3, 134|2, 1|234.
a(5) = 5: 1234|5, 1235|4, 1245|3, 1345|2, 1|2345.
a(6) = 66: 12345|6, 12346|5, 12356|4, 123|45|6, 123|46|5, 123|4|56, 12456|3, 124|35|6, 124|36|5, 124|3|56, 125|34|6, 12|345|6, 126|34|5, 12|346|5, 125|36|4, 125|3|46, 126|35|4, 12|356|4, 126|3|45, 12|3|456, 13456|2, 134|25|6, 134|26|5, 134|2|56, 135|24|6, 13|245|6, 136|24|5, 13|246|5, 135|26|4, 135|2|46, 136|25|4, 13|256|4, 136|2|45, 13|2|456, 145|23|6, 14|235|6, 146|23|5, 14|236|5, 15|234|6, 1|23456, 16|234|5, 1|234|56, 156|23|4, 15|236|4, 16|235|4, 1|235|46, 1|236|45, 1|23|456, 145|26|3, 145|2|36, 146|25|3, 14|256|3, 146|2|35, 14|2|356, 156|24|3, 15|246|3, 16|245|3, 1|245|36, 1|246|35, 1|24|356, 156|2|34, 15|2|346, 1|256|34, 1|25|346, 16|2|345, 1|26|345.
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MAPLE
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b:= proc(n, i, k) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, 1, `if`(i<2, 0, b(n, i-1, `if`(i=k, 0, k)))+
`if`(i=k, 0, b(n-i, min(n-i, i-1), k)*binomial(n, i))))
end:
a:= n-> b(n$2, 0)-b(n$2, 1):
seq(a(n), n=0..29);
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MATHEMATICA
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b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n == 0, 1, If[i < 2, 0, b[n, i - 1, If[i == k, 0, k]]] + If[i == k, 0, b[n - i, Min[n - i, i - 1], k] Binomial[n, i]]]];
a[n_] := b[n, n, 0] - b[n, n, 1];
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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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