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A318156 Expansion of (1/(1 - x)) * Sum_{k>=1} x^(k*(2*k-1)) / Product_{j=1..2*k-1} (1 - x^j).

%I A318156 #14 Jul 20 2021 03:29:42
%S A318156 0,1,2,3,4,5,7,9,12,16,21,27,35,44,55,69,85,104,127,154,186,224,268,
%T A318156 320,381,452,534,630,741,869,1017,1187,1382,1606,1862,2155,2489,2869,
%U A318156 3301,3792,4349,4979,5692,6497,7405,8429,9581,10876,12331,13963,15792,17840,20131,22691
%N A318156 Expansion of (1/(1 - x)) * Sum_{k>=1} x^(k*(2*k-1)) / Product_{j=1..2*k-1} (1 - x^j).
%C A318156 Partial sums of A067659.
%H A318156 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 A318156 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A318156 a(n) = A036469(n) - A318155(n).
%F A318156 a(n) = A318155(n) - A078616(n).
%F A318156 a(n) ~ exp(Pi*sqrt(n/3)) * 3^(1/4) / (4*Pi*n^(1/4)). - _Vaclav Kotesovec_, Aug 20 2018
%e A318156 From _Gus Wiseman_, Jul 18 2021: (Start)
%e A318156 Also the number of strict integer partitions of 2n+1 of even length with exactly one odd part. For example, the a(1) = 1 through a(8) = 12 partitions are:
%e A318156   (2,1)  (3,2)  (4,3)  (5,4)  (6,5)   (7,6)      (8,7)      (9,8)
%e A318156          (4,1)  (5,2)  (6,3)  (7,4)   (8,5)      (9,6)      (10,7)
%e A318156                 (6,1)  (7,2)  (8,3)   (9,4)      (10,5)     (11,6)
%e A318156                        (8,1)  (9,2)   (10,3)     (11,4)     (12,5)
%e A318156                               (10,1)  (11,2)     (12,3)     (13,4)
%e A318156                                       (12,1)     (13,2)     (14,3)
%e A318156                                       (6,4,2,1)  (14,1)     (15,2)
%e A318156                                                  (6,4,3,2)  (16,1)
%e A318156                                                  (8,4,2,1)  (6,5,4,2)
%e A318156                                                             (8,4,3,2)
%e A318156                                                             (8,6,2,1)
%e A318156                                                             (10,4,2,1)
%e A318156 Also the number of integer partitions of 2n+1 covering an initial interval and having even maximum and alternating sum 1.
%e A318156 (End)
%p A318156 b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
%p A318156       `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
%p A318156     end:
%p A318156 a:= proc(n) option remember; b(n$2, 0)+`if`(n>0, a(n-1), 0) end:
%p A318156 seq(a(n), n=0..60);
%t A318156 nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k (2 k - 1))/Product[(1 - x^j), {j, 1, 2 k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]
%t A318156 nmax = 53; CoefficientList[Series[(QPochhammer[-x, x] - QPochhammer[x])/(2 (1 - x)), {x, 0, nmax}], x]
%t A318156 Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&EvenQ[Length[#]]&&Count[#,_?OddQ]==1&]],{n,0,15}] (* _Gus Wiseman_, Jul 18 2021 *)
%Y A318156 Partial sums of A067659.
%Y A318156 The following relate to strict integer partitions of 2n+1 of even length with exactly one odd part.
%Y A318156 - Allowing any length gives A036469.
%Y A318156 - The non-strict version is A306145.
%Y A318156 - The version for odd length is A318155 (non-strict: A304620).
%Y A318156 - Allowing any number of odd parts gives A343942 (odd bisection of A067661).
%Y A318156 A000041 counts partitions.
%Y A318156 A027187 counts partitions of even length (strict: A067661).
%Y A318156 A078408 counts strict partitions of 2n+1 (odd bisection of A000009).
%Y A318156 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A318156 Cf. A000070, A030229, A035294, A058696, A078616, A087447, A152146, A236559, A343941, A344611, A344739.
%K A318156 nonn
%O A318156 0,3
%A A318156 _Ilya Gutkovskiy_, Aug 19 2018