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%I A346634 #11 Aug 05 2021 17:43:02
%S A346634 1,1,1,2,4,6,9,14,19,27,38,52,71,96,128,170,224,293,380,491,630,805,
%T A346634 1024,1295,1632,2048,2560,3189,3958,4896,6038,7424,9100,11125,13565,
%U A346634 16496,20013,24223,29250,35244,42378,50849,60896,72789,86841,103424,122960,145937
%N A346634 Number of strict odd-length integer partitions of 2n + 1.
%H A346634 Alois P. Heinz, <a href="/A346634/b346634.txt">Table of n, a(n) for n = 0..2000</a>
%e A346634 The a(0) = 1 through a(7) = 14 partitions:
%e A346634   (1)  (3)  (5)  (7)      (9)      (11)     (13)      (15)
%e A346634                  (4,2,1)  (4,3,2)  (5,4,2)  (6,4,3)   (6,5,4)
%e A346634                           (5,3,1)  (6,3,2)  (6,5,2)   (7,5,3)
%e A346634                           (6,2,1)  (6,4,1)  (7,4,2)   (7,6,2)
%e A346634                                    (7,3,1)  (7,5,1)   (8,4,3)
%e A346634                                    (8,2,1)  (8,3,2)   (8,5,2)
%e A346634                                             (8,4,1)   (8,6,1)
%e A346634                                             (9,3,1)   (9,4,2)
%e A346634                                             (10,2,1)  (9,5,1)
%e A346634                                                       (10,3,2)
%e A346634                                                       (10,4,1)
%e A346634                                                       (11,3,1)
%e A346634                                                       (12,2,1)
%e A346634                                                       (5,4,3,2,1)
%p A346634 b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
%p A346634      `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
%p A346634     end:
%p A346634 a:= n-> b(2*n+1$2, 0):
%p A346634 seq(a(n), n=0..80);  # _Alois P. Heinz_, Aug 05 2021
%t A346634 Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,15}]
%Y A346634 Odd bisection of A067659, which is ranked by A030059.
%Y A346634 The even version is the even bisection of A067661.
%Y A346634 The case of all odd parts is counted by A069911 (non-strict: A078408).
%Y A346634 The non-strict version is A160786, ranked by A340931.
%Y A346634 The non-strict even version is A236913, ranked by A340784.
%Y A346634 The even-length version is A343942 (non-strict: A236914).
%Y A346634 The even-sum version is A344650 (non-strict: A236559 or A344611).
%Y A346634 A000009 counts partitions with all odd parts, ranked by A066208.
%Y A346634 A000009 counts strict partitions, ranked by A005117.
%Y A346634 A027193 counts odd-length partitions, ranked by A026424.
%Y A346634 A027193 counts odd-maximum partitions, ranked by A244991.
%Y A346634 A058695 counts partitions of odd numbers, ranked by A300063.
%Y A346634 A340385 counts partitions with odd length and maximum, ranked by A340386.
%Y A346634 Other cases of odd length:
%Y A346634 - A024429 set partitions
%Y A346634 - A089677 ordered set partitions
%Y A346634 - A166444 compositions
%Y A346634 - A174726 ordered factorizations
%Y A346634 - A332304 strict compositions
%Y A346634 - A339890 factorizations
%Y A346634 Cf. A000700, A008289, A047993, A072233, A106529, A168659, A218171, A340102, A340604, A340607.
%K A346634 nonn
%O A346634 0,4
%A A346634 _Gus Wiseman_, Aug 01 2021
%E A346634 More terms from _Alois P. Heinz_, Aug 05 2021

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