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A120452
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Number of partitions of n-1 boys and one girl with no couple.
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45
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1, 1, 3, 5, 9, 14, 23, 34, 52, 75, 109, 153, 216, 296, 407, 549, 739, 981, 1300, 1702, 2224, 2879, 3716, 4761, 6083, 7721, 9774, 12306, 15450, 19307, 24064, 29867, 36978, 45614, 56130, 68846, 84250, 102793, 125148, 151955, 184123, 222553, 268482
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OFFSET
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1,3
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COMMENTS
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Also the number of:
- integer partitions of 2n with reverse-alternating sum 2;
- reversed integer partitions of 2n with alternating sum 2;
- integer partitions of 2n with exactly two odd parts, one of which is the greatest;
- odd-length integer partitions of 2n whose conjugate partition has exactly two odd parts.
Note that integer partitions of 2n with alternating or reverse-alternating sum 0 are counted by A000041, ranked by A000290.
(End)
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LINKS
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FORMULA
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a(n) = A000070(n-2) + A002865(n-1). - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Aug 15 2006
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)) * (1 - 37*Pi/(24*sqrt(6*n))). - Vaclav Kotesovec, Oct 25 2016
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EXAMPLE
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n=5:
If partitions have no pair "o*", then a(5)=9 ("o" means a boy, "*" means a girl): {o, o, o, o, *}, {o, o, *, oo}, {*, oo, oo}, {o, *, ooo}, {o, o, oo*}, {oo, oo*}, {*, oooo}, {o, ooo*}, {oooo*}.
The a(1) = 1 through a(6) = 14 partitions of 2n with reverse-alternating sum 2:
(2) (211) (222) (332) (442) (552)
(321) (431) (541) (651)
(21111) (22211) (22222) (33222)
(32111) (32221) (33321)
(2111111) (33211) (43221)
(43111) (44211)
(2221111) (54111)
(3211111) (2222211)
(211111111) (3222111)
(3321111)
(4311111)
(222111111)
(321111111)
(21111111111)
For example, the partition (43221) has reverse-alternating sum 1 - 2 + 2 - 3 + 4 = 2, so is counted under a(6).
The a(1) = 1 through a(6) = 14 partitions of 2n with exactly two odd parts, one of which is the greatest:
(11) (31) (33) (53) (55) (75)
(51) (71) (73) (93)
(321) (332) (91) (111)
(521) (532) (543)
(3221) (541) (552)
(721) (732)
(3322) (741)
(5221) (921)
(32221) (5322)
(5421)
(7221)
(33222)
(52221)
(322221)
(End)
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MATHEMATICA
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a[n_] := Total[PartitionsP[Range[0, n-3]]] + PartitionsP[n-1];
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CROSSREFS
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A000097 counts partitions of 2n with alternating sum 2.
A001700/A088218 appear to count compositions with reverse-alternating sum 2.
A344610 counts partitions of 2n by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A344741 counts partitions of 2n with reverse-alternating sum -2.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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More terms from Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Aug 15 2006
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STATUS
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approved
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