This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A120452 #24 Jun 11 2021 18:25:49
%S A120452 1,1,3,5,9,14,23,34,52,75,109,153,216,296,407,549,739,981,1300,1702,
%T A120452 2224,2879,3716,4761,6083,7721,9774,12306,15450,19307,24064,29867,
%U A120452 36978,45614,56130,68846,84250,102793,125148,151955,184123,222553,268482
%N A120452 Number of partitions of n-1 boys and one girl with no couple.
%C A120452 From _Gus Wiseman_, Jun 08 2021: (Start)
%C A120452 Also the number of:
%C A120452 - integer partitions of 2n with reverse-alternating sum 2;
%C A120452 - reversed integer partitions of 2n with alternating sum 2;
%C A120452 - integer partitions of 2n with exactly two odd parts, one of which is the greatest;
%C A120452 - odd-length integer partitions of 2n whose conjugate partition has exactly two odd parts.
%C A120452 Note that integer partitions of 2n with alternating or reverse-alternating sum 0 are counted by A000041, ranked by A000290.
%C A120452 (End)
%H A120452 Vaclav Kotesovec, <a href="/A120452/b120452.txt">Table of n, a(n) for n = 1..10000</a>
%F A120452 a(n) = A000070(n-2) + A002865(n-1). - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Aug 15 2006
%F A120452 a(n) = A000070(n-1) - A000041(n-2) = A000070(n-3) + A000041(n-1). - _Max Alekseyev_, Aug 23 2006
%F A120452 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
%e A120452 n=5:
%e A120452 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*}.
%e A120452 From _Gus Wiseman_, Jun 08 2021: (Start)
%e A120452 The a(1) = 1 through a(6) = 14 partitions of 2n with reverse-alternating sum 2:
%e A120452 (2) (211) (222) (332) (442) (552)
%e A120452 (321) (431) (541) (651)
%e A120452 (21111) (22211) (22222) (33222)
%e A120452 (32111) (32221) (33321)
%e A120452 (2111111) (33211) (43221)
%e A120452 (43111) (44211)
%e A120452 (2221111) (54111)
%e A120452 (3211111) (2222211)
%e A120452 (211111111) (3222111)
%e A120452 (3321111)
%e A120452 (4311111)
%e A120452 (222111111)
%e A120452 (321111111)
%e A120452 (21111111111)
%e A120452 For example, the partition (43221) has reverse-alternating sum 1 - 2 + 2 - 3 + 4 = 2, so is counted under a(6).
%e A120452 The a(1) = 1 through a(6) = 14 partitions of 2n with exactly two odd parts, one of which is the greatest:
%e A120452 (11) (31) (33) (53) (55) (75)
%e A120452 (51) (71) (73) (93)
%e A120452 (321) (332) (91) (111)
%e A120452 (521) (532) (543)
%e A120452 (3221) (541) (552)
%e A120452 (721) (732)
%e A120452 (3322) (741)
%e A120452 (5221) (921)
%e A120452 (32221) (5322)
%e A120452 (5421)
%e A120452 (7221)
%e A120452 (33222)
%e A120452 (52221)
%e A120452 (322221)
%e A120452 (End)
%t A120452 a[n_] := Total[PartitionsP[Range[0, n-3]]] + PartitionsP[n-1];
%t A120452 Array[a, 50] (* _Jean-François Alcover_, Jun 05 2021 *)
%Y A120452 Cf. A000041, A000070, A002865.
%Y A120452 A diagonal of A103919.
%Y A120452 A diagonal of A344612.
%Y A120452 A000097 counts partitions of 2n with alternating sum 2.
%Y A120452 A001700/A088218 appear to count compositions with reverse-alternating sum 2.
%Y A120452 A058696 counts partitions of 2n, ranked by A300061.
%Y A120452 A344610 counts partitions of 2n by sum and positive reverse-alternating sum.
%Y A120452 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A120452 A344741 counts partitions of 2n with reverse-alternating sum -2.
%Y A120452 Cf. A001250, A027187, A119899, A124754, A239830, A316524, A325535, A344618, A344651, A344739.
%K A120452 nonn,easy
%O A120452 1,3
%A A120452 _Yasutoshi Kohmoto_, Jul 20 2006
%E A120452 More terms from Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Aug 15 2006
%E A120452 More terms from _Max Alekseyev_, Aug 23 2006