OFFSET
1,3
COMMENTS
From Gus Wiseman, Jun 08 2021: (Start)
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)
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
FORMULA
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
EXAMPLE
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*}.
From Gus Wiseman, Jun 08 2021: (Start)
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)
MATHEMATICA
a[n_] := Total[PartitionsP[Range[0, n-3]]] + PartitionsP[n-1];
Array[a, 50] (* Jean-François Alcover, Jun 05 2021 *)
CROSSREFS
A diagonal of A103919.
A diagonal of A344612.
A000097 counts partitions of 2n with 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.
KEYWORD
nonn,easy
AUTHOR
Yasutoshi Kohmoto, Jul 20 2006
EXTENSIONS
More terms from Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Aug 15 2006
More terms from Max Alekseyev, Aug 23 2006
STATUS
approved