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A318155 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(k*(2*k+1)) / Product_{j=1..2*k} (1 - x^j).

%I A318155 #19 Mar 16 2024 11:21:50
%S A318155 1,1,1,2,3,5,7,10,13,17,22,28,35,44,55,68,84,103,126,153,185,223,268,
%T A318155 320,381,452,535,631,742,870,1018,1188,1383,1607,1863,2155,2489,2869,
%U A318155 3301,3792,4348,4978,5691,6496,7404,8428,9580,10875,12330,13962,15791,17840,20131,22691
%N A318155 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(k*(2*k+1)) / Product_{j=1..2*k} (1 - x^j).
%C A318155 Partial sums of A067661.
%C A318155 From _Gus Wiseman_, Jul 29 2021: (Start)
%C A318155 Also the number of strict integer partitions of 2n+1 of odd length with exactly one odd part. For example, the a(1) = 1 through a(7) = 10 partitions are:
%C A318155   (1)  (3)  (5)  (7)      (9)      (11)     (13)      (15)
%C A318155                  (4,2,1)  (4,3,2)  (5,4,2)  (6,4,3)   (6,5,4)
%C A318155                           (6,2,1)  (6,3,2)  (6,5,2)   (7,6,2)
%C A318155                                    (6,4,1)  (7,4,2)   (8,4,3)
%C A318155                                    (8,2,1)  (8,3,2)   (8,5,2)
%C A318155                                             (8,4,1)   (8,6,1)
%C A318155                                             (10,2,1)  (9,4,2)
%C A318155                                                       (10,3,2)
%C A318155                                                       (10,4,1)
%C A318155                                                       (12,2,1)
%C A318155 The following relate to these partitions:
%C A318155 - Not requiring odd length gives A036469.
%C A318155 - The non-strict version is A304620.
%C A318155 - The version for even instead of odd length is A318156.
%C A318155 - Allowing any number of odd parts gives A346634 (bisection of A067659).
%C A318155 (End)
%H A318155 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 16.4.2 "Partitions into distinct parts", page 350.
%H A318155 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A318155 a(n) = A036469(n) - A318156(n).
%F A318155 a(n) = A318156(n) + A078616(n).
%F A318155 a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - _Vaclav Kotesovec_, Aug 20 2018
%t A318155 nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k (2 k + 1))/Product[(1 - x^j), {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]
%t A318155 nmax = 53; CoefficientList[Series[(QPochhammer[-x, x] + QPochhammer[x])/(2 (1 - x)), {x, 0, nmax}], x]
%t A318155 Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&OddQ[Length[#]]&&Count[#,_?OddQ]==1&]],{n,0,15}] (* _Gus Wiseman_, Jul 29 2021 *)
%Y A318155 Cf. A036469, A078616, A304620, A318156.
%Y A318155 First differences are A067661 (non-strict: A027187, odd bisection: A343942).
%Y A318155 A000041 counts partitions.
%Y A318155 A000070 counts partitions with alternating sum 1.
%Y A318155 A078408 counts strict partitions of 2n+1 (odd bisection of A000009).
%Y A318155 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A318155 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A318155 Cf. A027193, A035294, A067659, A087447, A236559, A236914, A239829, A306145, A344611, A344739, A346634.
%K A318155 nonn
%O A318155 0,4
%A A318155 _Ilya Gutkovskiy_, Aug 19 2018